Webassign Calculus 1 Answers

9/22/2020 1. HW # 0 Fall 20 - MATH section 1 [0.32/0.43 Points] MY NOTES DETAILS PREVIOUS ANSWERS ASK YOUR TEACHER 1, Fall 2020 | WebAssign SCALC8 1.1.004. PRACTICE ANOTHER The graphs of f and g are given. (a) State the values of and = 2 2 g(3) = 4 4 (b) For what values of x is f(x) = g(x)? (Enter your answers as a comma-separated list.) x= $$−2,2 (c) Estimate the solutions of the equation f(x) = −1. (Enter your answers as a comma-separated list.) x= $$−3,4 (d) On what interval is f decreasing? (Enter your answer using interval notation.) $$[0,4] (e) State the domain and range of f. (Enter your answers in interval notation.) $$[−4,4] domain $$[−2,3] range (f) State the domain and range of g. (Enter your answers in interval notation.) $$[−4,3] domain 3/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign range $$[2.5,4] Solution or Explanation Click to View Solution Enhanced Feedback Please try again. For finding function values on a graph, remember to first locate the vertical line corresponding to the given x-value, then locate the y-value of the point at which it intersects the function. For finding all x-values that correspond to a given function value, remember to first locate the horizontal line corresponding to the given function value, then locate the x-values of any points at which it intersects the function. For finding the interval on which f is decreasing, remember that f is decreasing if f(x1) > f(x2) for any x1, x2 such that x1 < x2 on the interval. The domain of a function is the set of inputs on which the function is defined, whereas the range is the set of all outputs. 4/33 9/22/2020 2. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.28/0.43 Points] DETAILS PREVIOUS ANSWERS SCALC8 1.1.009. MY NOTES ASK YOUR TEACHER Consider the following graph. Determine whether the curve is the graph of a function of x. Yes, it is a function. No, it is not a function. If it is, state the domain and range of the function. (Enter your answers using interval notation. If it is not a function, enter DNE in all blanks.) $$[−3,−2]∪[−2,2] domain $$[−3,−2]∪[−1,3] range Solution or Explanation Click to View Solution Enhanced Feedback Please try again. Remember, a function is a rule that assigns to each element x in the domain exactly one element, called f(x), in the range. The graph of f is the set of points (x, y) in the plane such that y = f(x). 5/33 9/22/2020 3. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.35/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER If PREVIOUS ANSWERS SCALC8 1.1.025. PRACTICE ANOTHER find the following. f(3) = 27 27 f(−3) = 33 33 $$3a2−a+3 f( a ) = $$3a2+a+3 f(−a) = $$3a2+5a+5 f(a + 1) = $$6a2−2a+6 2f(a) = $$12a2−2a+3 f(2a) = $$3a4−a2+3 f( a 2 ) = $$6a2−2a+6 [f(a)]2 = $$3a2+6ah+3h2−a−h+3 f( a + h) = Solution or Explanation Click to View Solution Enhanced Feedback Please try again. To evaluate f for a specific term, replace each x by that term in the expression for f(x). To ensure correct simplification, start by surrounding each instance of the term with parentheses. 6/33 9/22/2020 4. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.1.027. PRACTICE ANOTHER Evaluate the difference quotient for the given function. Simplify your answer. $$−h−2x+4. Solution or Explanation Click to View Solution Enhanced Feedback Please try again. To evaluate f(a + h), replace x by (a + h) in the expression for f(x). Substitute this into the given expression and simplify. 5. [0/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.1.030. PRACTICE ANOTHER Evaluate the difference quotient for the given function. Simplify your answer. f( x ) = x+7 , x+3 f( x ) − f( 1 ) x−1 $$−1h+4 Solution or Explanation Click to View Solution 7/33 9/22/2020 6. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0/0.43 Points] MY NOTES DETAILS PREVIOUS ANSWERS ASK YOUR TEACHER SCALC8 1.1.051. PRACTICE ANOTHER Find an expression for the function whose graph is the given curve. (Assume that the points are in the form (x, f(x)).) The line segment joining the points f( x ) = $$(x+3)(x+7) Find the domain of the function. (Enter your answer using interval notation.) $$(−∞,∞) Solution or Explanation Click to View Solution 7. [0.43/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.1.061. PRACTICE ANOTHER Find a formula for the described function. An open rectangular box with volume 4 m3 has a square base. Express the surface area SA of the box as a function of the length of a side of the base, x. SA = $$x2+(16x) m2 State the domain of SA. (Enter your answer in interval notation.) $$(0,∞) Solution or Explanation Click to View Solution 8/33 9/22/2020 8. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.43/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.1.063. PRACTICE ANOTHER A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x. V( x) = $$4x3−64x2+240x Solution or Explanation Click to View Solution 9/33 9/22/2020 9. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.22/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.1.513.XP.MI. PRACTICE ANOTHER Find the domain of the function. (Enter your answer in interval notation.) $$(17,∞) Sketch the graph of the function. Solution or Explanation Click to View Solution 10/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign 11/33 9/22/2020 10. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.36/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.3.002. PRACTICE ANOTHER Explain how each graph is obtained from the graph of y = f(x). (a) y = f( x ) + 3 Shift the graph 3 units to the left. Shift the graph 3 units to the right. Stretch the graph horizontally and vertically by a factor of 3. Shift the graph 3 units upward. Shift the graph 3 units downward. (b) y = f( x + 3 ) Shift the graph 3 units to the right. Shift the graph 3 units to the left. Shift the graph 3 units downward. Shift the graph 3 units upward. Stretch the graph horizontally and vertically by a factor of 3. (c) y = 3 f( x ) Stretch the graph vertically by a factor of 3. Shift the graph 3 units upward. Shift the graph 3 units to the left. Stretch the graph horizontally and vertically by a factor of 3. Shrink the graph horizontally by a factor of 3. (d) y = f(3x) Shrink the graph horizontally by a factor of 3. Shift the graph 3 units upward. Stretch the graph horizontally and vertically by a factor of 3. Stretch the graph vertically by a factor of 3. Shift the graph 3 units to the left. (e) y = − f( x ) − 1 12/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign First reflect the graph about the x-axis, and then shift it 1 unit left. First reflect the graph about the x-axis, and then shift it 1 unit upward. First reflect the graph about the y-axis, and then shift it 1 unit upward. First reflect the graph about the x-axis, and then shift it 1 unit downward. First reflect the graph about the y-axis, and then shift it 1 unit downward. (f) y = 3f 1 x 3 Shrink the graph horizontally by a factor of 3. Shrink the graph horizontally and vertically by a factor of 3. Stretch the graph horizontally and vertically by a factor of 3. Stretch the graph vertically by a factor of 3. Stretch the graph horizontally by a factor of 3. Solution or Explanation (a) To obtain the graph of y = f(x) + 3 from the graph of y = f(x), shift the graph 3 units upward. (b) To obtain the graph of y = f(x + 3) from the graph of y = f(x), shift the graph 3 units to the left. (c) To obtain the graph of y = 3f(x) from the graph of y = f(x), stretch the graph vertically by a factor of 3. (d) To obtain the graph of y = f(3x) from the graph of y = f(x), shrink the graph horizontally by a factor of 3. (e) To obtain the graph of y = −f(x) − 1 from the graph of y = f(x), first reflect the graph about the x-axis, and then shift it 1 unit downward. (f) To obtain the graph of y = 3f 1 x 3 from the graph of y = f(x), stretch the graph horizontally and vertically by a factor of 3. 13/33 9/22/2020 11. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.43/0.43 Points] DETAILS PREVIOUS ANSWERS SCALC8 1.3.003. MY NOTES ASK YOUR TEACHER The graph of y = f(x) is given. Match each equation with its graph. (a) y = f(x − 4) 1 2 3 4 5 (b) y = f( x ) + 3 1 2 3 4 5 (c) y= 1 f( x ) 3 1 2 3 4 5 (d) y = −f(x + 4) 14/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign 1 2 3 4 5 (e) y = 2f(x + 6) 1 2 3 4 5 Solution or Explanation Click to View Solution 15/33 9/22/2020 12. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.43/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.3.015. PRACTICE ANOTHER Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y = x 2 − 4x + 8 Solution or Explanation y = x2 − 4x + 8 = (x2 − 4x + 4) + 4 = (x − 2)2 + 4: Start with the graph of y = x2, shift 2 units to the right, and then shift upward 4 units. 16/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign 17/33 9/22/2020 13. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.35/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.3.031. PRACTICE ANOTHER Find each of the following functions and state their domains. (Enter the domains in interval notation.) (a) $$x3+12x2−3 f+g= $$(−∞,∞) domain (b) f−g $$x3−2x2+3 f−g= $$(−∞,∞) domain (c) fg $$7x5+35x4−3x3−15x2 fg = $$(−∞,∞) domain (d) f/ g $$x3+5x27x2−3 f/g = $$(−√37,√37) domain 18/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign Solution or Explanation Click to View Solution 19/33 9/22/2020 14. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.05/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS SCALC8 1.3.037. PRACTICE ANOTHER Find the functions and their domains. (Enter the domains in interval notation.) (a) (f ∘g f $$(2x2+38x+293)x2+19x+34 ∘ g)(x) = $$(0,∞) domain (b) g ∘f $$(x2+17x+1)(x2+2x+1) ∘ f)(x) = (g $$(0,∞) domain (c) (f f ∘f $$(18x2+87x+102)3x2+27x+42 ∘ f)(x) = $$(0,∞) domain (d) g ∘g $$(x2+17x+1)x2+2x+1 (g ∘ g)(x) = $$(0,∞) domain 20/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign Solution or Explanation Click to View Solution 15. [0.15/0.43 Points] DETAILS PREVIOUS ANSWERS SCALC8 1.3.053. MY NOTES ASK YOUR TEACHER Use the given graphs of f and g to evaluate each expression, or if the expression is undefined, enter UNDEFINED. (a) f(g(2)) 4 (b) 4 g(f(0)) 2 3 ∘ g)(0) (c) -2 (f (d) (g 0 2 (e) UNDEFINED (g -1 (f) -2 ∘ f)(6) ∘ g)(−2) 4 (f ∘ f)(4) -2 Solution or Explanation Click to View Solution Enhanced Feedback Please try again. Remember that given two functions f and g, the composite function f ∘g is defined by (f ∘ g)(x) = f(g(x)). To evaluate f(g(x)) using the graph, first find g(x) using the graph. Then, using g(x) as the input, find the corresponding function value of f from the graph. 21/33 9/22/2020 16. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.12/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS WEBASSIGNCRA1 8.1.001. PRACTICE ANOTHER Let f(x) = 6 − 2−x + 1. Part 1 - Graph Graph y = f(x). Part 2 - Domain and range From your graph in part 1, state the domain and range of f. (Enter your answers using interval notation.) $$(−∞,∞) domain $$(−∞,∞) range Part 3 - One-to-one From your graph in part 1, explain why f is one-to-one. Since y = f(x) passes the Vertical Line Test, f is one-to-one. Since y = f(x) does not pass the Horizontal Line Test, f is one-to-one. Since y = f(x) does not pass the Vertical Line Test, f is one-to-one. Since y = f(x) passes the Horizontal Line Test, f is one-to-one. 22/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign Part 4 - f −1(x) Find a formula for f −1(x). $$log2(−x2+1) f −1(x) = Part 5 - Function composition Check your answer to part 4 using function composition. (f −1 (f ∘ f)(x) = (No Response) ∘ f −1)(x) = (No Response) Additional Materials Watching 23/33 9/22/2020 17. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [–/0.43 Points] MY NOTES DETAILS WEBASSIGNCRA1 8.1.007.TUT. ASK YOUR TEACHER PRACTICE ANOTHER Find the x- and y-intercepts of the exponential function below. x-intercept (x, y) = (No Response) y-intercept (x, y) = (No Response) Solution or Explanation Part 1 of 1 Use the graph to see where the function crosses the x- and y-axes. In the graph below, the x- and y-intercepts are indicated with points. Recall that the x-intercept is the point (a, 0) where the graph crosses the x-axis. Look at the graph and find the value of x when the graph crosses the x-axis. The graph crosses the x-axis at what value? x=1 Therefore, the x-intercept is located at the following point. 24/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign (x, y) = Now recall that the y-intercept is the point (0, b) where the graph crosses the y-axis. Look at the graph and find the value of y when the graph crosses the y-axis. The graph crosses the y-axis at what value? y = -7 Therefore, the y-intercept is located at the following point. (x, y) = Additional Materials Tutorial 25/33 9/22/2020 18. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS WEBASSIGNCRA1 8.1.012. PRACTICE ANOTHER Graph the given exponential function and then graph its inverse function on the same coordinate system. f (x) = 1 4 x Solution or Explanation Make a table of values for f (x). x f (x) −1 1 −1 = 4 4 0 1 0= 1 4 1 1 1 1 = 4 4 26/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign Plot these points and connect them with a smooth curve. Then, the graph of f −1(x) will be the reflection of this graph in the line y = x. To graph a function, make a table of ordered pairs. The inverse will be a reflection of the original graph. Across what line is it reflected? 27/33 9/22/2020 19. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [–/0.43 Points] DETAILS Find the domain of f(x) = WEBASSIGNCRA1 8.2.004. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER 2 − ex . (Enter your answer using interval notation.) (No Response) Find f −1(x). (No Response) Find the domain of f −1(x). (Enter your answer using interval notation.) (No Response) Verify that your functions are inverses using composition of functions. (No Response) Key: Answers will vary. This answer has not been graded yet. Solution or Explanation f( x ) = 2 − ex domain of f: 2 − ex ≥ 0 2 ≥ ex ex ≤ 2 x ≤ ln 2 2 − ex y = f(x) = x = 2− (−∞, ln 2] ey x 2 = 2 − ey ⇒ ey = 2 − x 2 y = ln(2 − x2) ⇒ f −1(x) = ln(2 − x2) domain of f −1 = range of f: lim f(x) = 2 x → −∞ f(ln 2) = 0 domain of f −1: 0, 2 f(f −1(x)) = f(ln(2 − x2)) = = f −1(f(x)) = f −1 2 − ex 2 2 − eln(2 − x ) = 2 − 2 + x2 = = ln 2 − 2 − ( 2 − x2 ) x 2 = |x | = x 2 − ex (since x ≥ 0) 2 = ln 2 − (2 − ex) = ln(2 − 2 + ex) = ln ex = x 28/33 9/22/2020 20. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS WEBASSIGNCRA1 8.2.009. PRACTICE ANOTHER State the asymptote of the logarithmic function. $$x=5 Solution or Explanation On a graph, a vertical asymptote exists at x = a if the y-values approach positive or negative infinity as x gets very close to a. Because logarithms are only defined on positive numbers, a vertical asymptote is determined by those values for which the argument of the logarithm is equal to 0. 7x = 0 Divide both sides by 7. x = 0 The vertical asymptote of a logarithmic function will be the line x = a, where the function value approaches a as y gets very large, or very small. What value is approached by this graph? Did you give the answer as an equation of a line? 29/33 9/22/2020 21. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0/0.43 Points] MY NOTES DETAILS PREVIOUS ANSWERS ASK YOUR TEACHER WEBASSIGNCRA1 8.2.012. PRACTICE ANOTHER Find the inverse function, f −1(x). f ( x ) = −2x f −1(x) = (No Response) , x< 2 0 Solution or Explanation Substitute y for f (x). y = −2x x = −2y −x = 2y log(−x) = log 2y log(−x) = y log 2 log(−x) log 2 f −1(x) Interchange x and y, and then solve for y. Isolate the exponential expression. Take the logarithm of both sides. Note: The log with any base b may be used. Apply the Logarithm Property of the Power of a Number. Divide by log 2 to isolate y. Replace y with f −1(x). = y Restrict the domain because the logarithm is undefined on non-positive numbers. = log(−x) ,x<0 log 2 First substitute y for f(x), and then switch the x and y variables in the equation. How does taking the logarithm (common or natural) of both sides of the equation help to isolate y? How is y related to the inverse function f −1(x)? 30/33 9/22/2020 22. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0/0.43 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS WEBASSIGNCRA1 8.2.014. PRACTICE ANOTHER Find the inverse function, f −1(x), of the logarithmic function. f (x) = 7 log6(x + 3) f −1(x) = $$1763x Solution or Explanation Recall the definition of the logarithm: For b > 0 and b ≠ 1, v = logb u is equivalent to u = bv. Substitute y for f (x). y = 7 log6(x + 3) Interchange x and y, and then solve for y. x = 7 log6(y + 3) Isolate the logarithm by dividing by 7. x 7 Use the definition of the logarithm to write 6x/7 6x/7 = log6(y + 3) = y+3 −3 = y the expression in exponential form. Solve for y. Replace y with f −1(x). f −1(x) = 6x/7 − 3 Replace f(x) with y, and then switch the variables x and y. How do you convert the equation from a logarithmic equation to an exponential equation in order to isolate y? How is this y and f−1(x) related? 31/33 9/22/2020 23. HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign [0.54/0.54 Points] MY NOTES DETAILS ASK YOUR TEACHER PREVIOUS ANSWERS WEBASSIGNCRA1 8.2.015. PRACTICE ANOTHER Graph the logarithmic function using transformations of y = f (x) = log2 x. f (x) = 5 log2(x − 1) + 4 Solution or Explanation Begin with the graph of y = log2 x and use transformations. To graph y = log2(x − 1), shift the graph of y = log2 x to the right 1 unit. From there, to graph f (x) = 5 log2(x − 1) + 4, take the graph of y = log2(x − 1), stretch it vertically by a factor of 5, and shift it upward 4 units. 32/33 9/22/2020 HW # 0 Fall 20 - MATH 151, section 11, Fall 2020 | WebAssign Home My Assignments Request Extension 33/33