Unit 2 Library of Functions

Unit 2 Library of Functions ● Square Root ○ Parent function: 𝑓(𝑥) = √𝑥 ○ 3 common points: ■ (0,0) ■ (1,1) ■ (4,2) ○ Domain: [0,∞) ○ Range: [0,∞) ● Cube Root 3 ○ Parent function: 𝑓(𝑥) = √𝑥 ○ 3 common points: ■ (-1,1) ■ (0,0) ■ (3,3) ○ Domain: (-∞,∞) ○ Range: (-∞,∞) ● Constant ○ Parent function: f(x) = b where b is a constant ○ 3 common points: ■ (-1,b) ■ (0,b) ■ (1,b) ○ Domain: (-∞,∞) ○ Range: {y| y = b} [b] ● Absolute ○ Parent function: 𝑓(𝑥) = |𝑥| ○ 3 Common Points ■ (-1,1) ■ (0,0) ■ (1,1) ○ Domain: (-∞,∞) ○ Range: [0,∞) ● Linear/Identity ○ Parent Function: 𝑓(𝑥) = 𝑥 ○ 3 Common Points: ■ (-1,1) ■ (0,0) ■ (1,1) ○ Domain: (-∞,∞) ○ Range: (-∞,∞) ● Square/Quadratic ○ Parent Function: 𝑓(𝑥) = 𝑥 2 ○ 3 Common Points: ■ (-1,1) ■ (0,0) ■ (1,1) ○ Domain: (-∞,∞) ○ Range: [0,∞) ● Cubic ○ Parent Function:𝑓(𝑥) = 𝑥 3 ○ 3 Common Points: ■ (-1,-1) ■ (0,0) ■ (1,1) ○ Domain: (-∞,∞) ○ Range: (-∞,∞) ● Rational/Reciprocal 1 ○ Parent Function: 𝑓(𝑥) = 𝑥 ○ 5 Common points ■ (-1 ,1) ■ (1, 1) ■ (2 ,½) ■ (-2, ½) ■ (½, 2) ○ Domain: (-∞,0) ∪ (0,∞) ○ Range:(-∞,0) ∪ (0,∞) ● Greatest Integer ○ Parent Function: [[x]] = int(x) ○ Common Points: ■ (0,1) 0) ■ (1,2) 1) ○ Domain: (-∞,∞) ○ Range: {y| y = z} Transformations ● Example Parent function: 𝑓(𝑥) = 𝑥 2 ○ 𝑦 = 𝑎(𝑏𝑥 + 𝑐)2 + 𝑑 ● Vertical Stretch: a > 1 ● Vertical Compression: 0 < a < 1 ● Reflection across x-axis: -a ● Horizontal Compression: b > 1 ● Horizontal Stretch: 0 < b < 1 ● Reflection across y-axis: -b ● Shifts Left: +c ● Shifts Right: -c ● Shifts Up: +d ● Shifts Down: -d Polynomial Functions ● f(x) = anxn + an-1xn-1 + . . . + a1x + a0 ○ The a’s are the coefficients. ○ n’s are exponents (all positive integers and 0) ● How to determine if a function is polynomial: ○ If there is a +, −, or × sign in the exponent, it is a polynomial.. ○ If there is a division sign, then it is a rational function. ○ It must have a domain of all real numbers. ● Let’s look at some examples: ○ f(x) = 2 - 3x4 ■ This is polynomial because there is a degree of 4. ○ h(x) = 0 ■ This is a polynomial because there is no degree. ○ g(x) = √𝑥 ■ You can rewrite this function as x1/2 ● There is a fraction in the exponent. 𝑥 2 −2 ○ f(x) = 𝑥 3 −1 ■ The numerator and denominator themselves are polynomials. However, as a whole, it is a rational function, because it doesn’t have a domain of all real numbers ● If the x - value was zero it would be undefined. ● Power functions ○ A power function is also known as a monomial, an expression that is made up of only one term. ○ f(x) = axn ● Properties of power functions when n is even: ○ Symmetry: y-axis ○ Domain: (-∞,∞) ○ Common Points: (-1,a) (0,0) (1,a) ● Properties of power functions when n is odd: ○ Symmetry: Origin ○ Domain: (-∞,∞) ○ Common Points: (-1,-a) (0,0) (1,a) ● Roots, Zeroes, and Intercepts: ○ Zero of a function: x-value that make the y-value =0 ○ The root of a function: The solutions of the polynomial equations when the function equals zero. ○ X-intercepts of a function: These are the points where the graph crosses the xaxis. ● Let’s Practice! ○ What are the roots of f(x) = x2 + x - 6 ? ■ Step 1: Factor ● f(x) = (x+3)(x-2) ■ Step 2: set ‘x+3’ and ‘x-2’ equal to zero and solve for x ● x + 3 = 0 => x = -3 ● x - 2 = 0 => x = 2 ■ Answer: The roots of the function are -3 and 2 ○ Find a polynomial of degree 3 whose zeros are -3,5,-7 ■ Degree 3 means that 3 is the term with the largest power ■ Answer: f(x) = (x+3)(x+7)(x-5) ● Multiplicity ○ This is when the same factor occurs more than once. ○ Example 1: f(x) = (x-2)(x+3)4(x-5)7 ■ What are the roots? How many times do they occur? ● The roots are 2,-3, and 5 ○ 2 occurs 1 time ○ -3 occurs 4 times ○ -5 occurs 7 times ● If you add how many times each of the 3 roots occur, you will find the degree of the function. ○ For this function, it is 12 ○ Graphing: ■ When the multiplicity is even, it bounces off the x-axis. ■ When the multiplicity is odd, the graph crosses the xaxis. ○ Example 2: f(x) = x2(x-2) ■ Roots? 2 and 0 ■ Multiplicity ● The multiplicity of 2 is 1, so it crosses the xaxis. ● The multiplicity of 0 is 2, so it bounces off the x-axis. Interval (-∞,0) (0,2) (2,∞) Sample Point x = -1 x=1 x=3 Function Value f (-1) = -3 f(1) = -1 f(3) = 9 Above or below? Below, the ‘y’ value below above (x-axis) is negative Point on graph (-1,-3) (1,-1) (3,9) ● Turning Points ○ If f is a degree n polynomial function, then f has n-1 turning points at most. ○ Turning points = degree - 1 ● End Behavior ○ If x is a really large number, the graph of the polynomial will look like the graph of the power function. ● Let's practice all these with these new concepts: ○ Example 3: f(x) = x3 + x2 - 12x ■ Y-intercept ● Step 1: substitute 0 in for ‘x’ ○ f(0) = 03 + 02 -12(0) ● Answer: (0,0) ■ X-intercepts ● Step 1: Factor out a what is common ○ All the terms have x in common, so you can bring that out ○ f(x) = x (x2 + x - 12) ● Step 2: Factor what is in the parentheses further ○ f(x) = x (x + 4) (x - 3) ● Answers: ○ (0,0) root: 0 multiplicity: 1 ○ ((-4,0) root: -4 multiplicity: 1 ○ (3,0) root: 3 multiplicity: 1 ■ Turning points ● Step 1: determine the degree the of the polynomial ○ Degree = 3 ● Step 2: subtract 1 from the degree ○ 3-1=2 ● Answer: 2 ■ End behavior ● The power function is y = x3 Real Zeros of a Polynomial Function ● Theorem: If the polynomial functions are denoted by f(x) and g(x), and if g(x) is not the zero polynomial, then there are special polynomial functions such that: ○ 𝑓(𝑥) 𝑟(𝑥) = 𝑞(𝑥) + 𝑔(𝑥) or 𝑔(𝑥) ○ Example 1: 𝑓(𝑥) = 𝑓(𝑥) = 𝑞(𝑥)𝑔(𝑥) + 𝑟(𝑥) 𝑥 2 +𝑥−12 𝑥−4 ● Remainder Theorem: Let f be a polynomial function. If f(x) is divided by x-c, then the remainder is f(c) ○ Example 2: Find the remainder using the remainder the remainder theorem from 𝐹(𝑥) = 𝑥 2 +𝑥−12 𝑥−4 ● Factor Theorem: Let f be a polynomial function. Then x-c is a factor of f (x) if and only if f (c) = 0 ○ What does that mean? 1. If f (c) = 0, then x-c is a factor of f (x) 2. If x - c is a factor of f (x), then f (c) = 0 ○ Example 3: ● Real Zeros ○ Theorem: A polynomial function may not have more real zeros than the highest degree. ○ Descartes’ rule of signs: ■ Let f denote a polynomial function written in standard form: ● The number of positive real zeros of f either equals the number of variations in the sign of the coefficients of f(x) or else equals that number less and even integer. ● The number of negative real zeros of f either equals the number of variations in the sign of the nonzero coefficients of f (-x) or else equals that number less an even integer. ○ Example 4: Discuss the real zeros of 𝑓(𝑥) = 3𝑥 6 − 4𝑥 4 + 3𝑥 3 + 2𝑥 2 − 𝑥 − 3 ■ Zeros: maximum 6 because that is the degree ■ Positive real numbers/sign changes: 3 or 1 ● Only change signs of odd degree terms ■ Negative Real: 3 or 1 ■ Imaginary: o or 2 or 4 ● Rational Zeros Theorem: ○ Let f be a polynomial function of degree 1 or higher of the form ■ 𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 +. . . +𝑎1 𝑥 + 𝑎0 𝑝 ○ Where each coefficient is an integer. If 𝑞, in lowest terms, is a rational zero of f, then p must be a factor of a0(Constant), and q must be a factor of an ○ Example 5: List the potential rational zeros of 𝑓(𝑥) = 2𝑥 3 + 11𝑥 2 − 7𝑥 − 6 ■ Degree tells you number of factors ■ p: 2,3,1,6 ■ q: 1,2 ■ 𝑝 𝑞 1 3 = ±(1,2,3,6, 2, 2) ● 12 numbers because it is + or - ● Theorem: Each polynomial function (with real coefficients) can be individually factored into the product of linear factors and/or square factors. ○ Linear ex; x+2 ○ Irreductible ex; x2-2 ● Corollary: A polynomial function (with real coefficients) of odd degree has at least one real zero. ○ Why? ■ Because it crosses the x-axis ■ It has an odd number of zeros ● Intermediate Value Theorem: Let f indicate a polynomial function. If a < b and if f (a) amd f(b) are of opposite sign, then there is at least one zero between a and b. ○ Example 6: Show that f(x) = x5 - x3 -1 has a zero between 1 and 2 Piecewise-defined Function ● One of the most common piecewise defined functions is.... ● Example 1: ○ What is f(0)? ■ 1 ○ What is f(1)? ■ ±2 ○ What is f(2)? ■ 4 ○ What is the domain of f(x)? ■ [ -1, ∞ ) ○ What is the range of f(x)? ■ ( 0, ∞ ) Composite Functions ● Notation: 𝑓(𝑔(𝑥)) 𝑜𝑟 ( 𝑓 ∘ 𝑔)(𝑥) ○ f “composed with” g ● Example: 𝑓(𝑥) = 2𝑥 2 − 3 𝑎𝑛𝑑 𝑔(𝑥) = 4𝑥 ○ Find 𝑓(𝑔(𝑥)) ■ 𝑓(4𝑥) = 2(4𝑥 2 ) − 3 ■ 32𝑥 2 − 3 ■ Inverse Functions ● To find the inverse of a function you switch the place of the x and y values. ○ Find the inverse of {(-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), (3,9)} ■ {(9,-3), (4,-2), (1,-1), (0,0), (1,1), (4,2), (9,3)} ● When the inverse of a function is also a function, then the original function is one-toone. ● Verifying two functions are inverses: ○ Example: f(x) = 2x+3, then f -1(x) = ½(x-3) ■ Part 1 ● f(½(x-3)) + 3 ● x-3+x ● x✓ ■ Part 2 ● ½(2x+3) - 3 ● x+3-3 ● x✓ ● The inverse of a function is its reflection across x=y. Rational Functions 𝑝(𝑥) ● 𝑅(𝑥) = 𝑞(𝑥) ○ p(x) = polynomial ○ q(x) = polynomial ○ R(x) = ratio of polynomial functions ● Finding the domain: ○ Problems occur in the domain of a rational function when q(x) = 0 ■ Makes the function undefined ● Example: ● Vertical Asymptote: As x approaches some number, c, the y-value approaches -∞ or ∞. (x=c is also a vertical asymptote.) ● Theorem: If r is a real denominator zero, a rational function in the lowest terms would have a vertical asymptote x= r. ● Vocabulary ○ Holes: Zeros of both the numerator and the denominator. ○ X-intercepts: Zero of the numerator only. ○ Vertical Asymptotes: Zero of the denominator only. ● End Behavior of Rational Functions: ○ The quotient of dividing serves as the end behavior. ● Horizontal Asymptotes ○ Theorem: If the numerator 's degree is less than the denominator 's degree, then the line y = 0 is a horizontal asymptote of its graph. 𝑥−12 ■ Example: 𝑓(𝑥) = 4𝑥 2+𝑥+1 ● Numerator has a degree of 1 and denominator has a degree of 2. ● Horizontal Asymptote: y = 0 ● It is a proper rational function. ○ Horizontal Asymptotes besides y=0 ■ Horizontal asymptotes also occur when the degree of the numerator and denominator are the same. 6𝑥 2 +𝑥+12 ● Example: 𝑓(𝑥) = 3𝑥 2−5𝑥−2 ○ 6/3 = 2 ○ Horizontal Asymptote: y = 2 ● Oblique/slant asymptotes arise when the numerator degree is greater than the denominator degree by 1. 3𝑥 4 −𝑥2 ○ Example: 𝑓(𝑥) = 𝑥 3 −𝑥 2+1 ● Graphing Rational Functions Checklist 1. Find the Domain of the function by setting the denominator ≠0 2. Locate the intercepts ■ Y: plug 0 in for x ■ X: set the numerator = 0 3. Test for symmetry- even(y-axis) or odd(origin) ■ Does R(-x) = R(x) or R(-x) = -R(x) 4. Find vertical asymptotes and/or holes by finding zeros of both the numerator and denominator 5. Find the end behavior (Horizontal or Oblique asymptotes using long division when necessary) 6. Find Points using intervals around x-intercepts, vertical asymptotes, and holes 7. Graph your points 8. Using properties of these to draw the graph by connecting points and approaching asymptotes Complex Zeros; Fundamental Theorem of Algebra ● Complex number: 𝑎 + 𝑏𝑖 ○ a -real number ○ b- ral number ○ 𝑖 = √−1 or 𝑖 2 = −1 ● Complex Numbers include both real and imaginary numbers. ● Fundamental Theorem of Algebra: Each complex polynomial function f(x) of degree n ≥ 1 has at least one complex zero. ○ Theorem: Every complex polynomial function f(x) of degree n ≥ 1 can be factored into n linear factors of the form 𝑓(𝑥) = 𝑎𝑛 (x-r1)(x-r2)...(x-rn) when an, r1, r2,..., rn are complex numbers. That is, every complex polynomial function of degree n ≥ 1 has exactly n zeros ■ n= degree, factors, or zeros ● Conjugate Pairs Theorem: Let f(x) be a polynomial whose coefficients are real numbers. If r=a+bi is a zero of f(x), then the complex conjugate 𝑟 = 𝑎 − 𝑏𝑖is also a zero f(x). ○ Imaginary zeros always come in pairs! ■ Example: If 2-3i is a zero what must also be a zero? ● 2+3i ● Corollary: ○ Polynomial f of odd degree with real coefficients has at least one real zero. ○ Example: If a polynomial f of degree 5 whose coefficients are real numbers has the real zeros 1, 5i, and 1+i, find the remaining two zeros: ■ 1-i ■ -5i Polynomial and Rational Inequalities 1. Solve the inequality for 0 a. Imagine f(x) > 0 b. If f(xP is a rational inequality, then express f(x) as the 1 ratio (fraction). 2. Find when f(x) = 0 a. Imagine a graph of the function f(x); the inequality means to find when the graph is above/below the x-axis, so we start by finding the x-intercepts. b. For a rational inequality, find when the numerator(x-intercepts) and denominator(vertical asymptotes) equal 0. 3. Create intervals around the zeroes of the function(numerator and denominator) 4. Test each interval to see which agree with the original relationship a. This is to find when the graph is above the x-axis (f(x)>0) b. Or when the graph is below the x-axis (f(x)<0) 5. Check to see if the ends of the intervals are included (hard or soft bracket) ● Example: x2 + 7x ≥ 0 Interval (-∞,-7) (-7,0) (0,∞) Sample Point x = -8 x = -1 x=1 Function Value y=8 y=-6 y=8 Above or Below? above below above Solution: (-∞,-7] ⋃ [0,∞)