Piecewise-Defined Functions #6 - Questions and Answers

Question 1 Find the x- and y-intercepts of the graph of the equation y = |3x-7|. Answer Step 1 Given y=|3x-7| Step 2 Substitute 0 for y and solve for x to obtain the x-intercept. |3x-7|=0 Remove the absolute value term. This creates a ± on the right side of the equation because |x| = ±x 3x-7=±0 3x-7=0 3x=7 x = 7/3 x-intercept in point form: (7/3, 0) Substitute 0 for x and solve for y to obtain the y-intercept. y=|3(0)-7| =|-7| =7 y-intercept in point form: (0,7) Question 2 Find the absolute maximum and absolute minimum values of f(x) = x⁴ - 25x² + 25 on the interval [-5,5]. Find the absolute maximum of f on the interval. Find the absolute minimum of f on the interval. Answer Step 1 Given, f(x) = x⁴ - 25x² + 25 and the interval [-5,5]. f(x) can be written as, f(x) = x⁴ - 25x² + 25 Step 2 Absolute maximum or absolute minimum values of a function lies at end points of the interval or at the points where first derivative is zero. So first we find the derivative of the given function, that is f'(x) = 4x³ - 25(2x) = 4x³ - 50x Step 3 Now finding the points where first derivative is zero, that is f'(x)=0 ⇒4x³ - 50x=0 ⇒2x(2x² - 25) = 0 2x = 0 or 2x² - 25= 0 x = 0 or x² = 25/2 x=0 or x = ±√(25/2) = ±5/√2 x = 0, ±5/√2 Step 4 Now computing the value of the given function at the end points of the given interval and at x =0 and ±5/√2 We get f(-5) = (-5)⁴ - 25(-5)² + 25 = 625 - 625 + 25= 25 f(5) = (5)⁴ - 25(5)² + 25 = 625 - 625 + 25 = 25 f(0) = (0)⁴ - 25(0)² + 25 = 25 f(5/√2) = (5/√2)⁴ - 25(5/√2)² + 25 = 625/4 - 25(25/2) + 25 = 625/4 - 625/2 + 25= (625-1250+100)/4= -525/4 = -131.25 f(-5/√2) = (-5/√2)⁴ - 25(-5/√2)² + 25 = 625/4 - 25(25/2) + 25 = 625/4 - 625/2 + 25= (625-1250+100)/4= -525/4= -131.25 Step 5 Hence, absolute maximum value of f is 25 occurs at the points x= -5, 0, 5. absolute minimum value of f is -131.25 occurs at the points x= ±5/√2. Question 3 Solve the absolute value equations for x. |2x-5|=|3x-7| Answer |2x-5|=|3x-7| First case: 2x-5=3x-7 3x-2x=7-5 x=2 Second case: 2x-5=-(3x-7) 2x-5=-3x+7 5x=12 x= 12/5 Question 4 The absolute value of a number is equal to the absoolute value of its opposite. Answer Calculation: Given: The absolute value of a number is equal to the absolute value of its opposite. Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative. The absolute number of a number is written as |a|. The symbol |a| is used to indicate the absolute value of a. |a| is a, If a is positive. |-a| is a, if a is negative. The equation |x| = a has two solutions x = a and x = -a because both numbers are at the distance a from 0. 1, A number and the opposite of the number always have the same absolute value. In general, taking the opposite of n changes the sign of n. For example, the opposite of 3 is -3. In general, taking the absolute value of n gives a number |n|, which is always positive. For example, |3| = 3 and |-3| = 3. 4, Since the opposite of 0 is 0 (which is neither positive nor negative), therefore -0 = 0. The number 0 is the only number which is its own opposite. Conclusion: The statement is true. Question 5 Define Absolute value. Answer Step 1 Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. The absolute value of a number is never negative. Step 2 For example, 6 is 6 away from zero, so the absolute value of 6 is 6 -6 is 6 away from zero, so the absolute value of -6 is 6 in other words it is the magnitude of a number, no negatives allowed.