University:
College Entrance Preparation
Course:
ACT Preparation (Mathematics)
Academic year:

2024
Views:

472

Pages:

4
Author:

Lamghari6kx

Algebra 2 Name: Date: Period: Function Inverses State if the given functions are inverses. 1. f(x) = 1/3x g(x) = 3x Yes 2. g(x) = -12 - 5n f(n) = (-5 - 24n)/5 No 3. f(x) = -2 + 1/4x g(x) = 16x + 8 No 4. f(x) = 4/7x - 16/7 g(x) = 7/4x + 4 Yes 5. f(n) = -(n + 1)^3 g(n) = 3√n + 1 No 6. f(n) = 2(n - 2)^3 g(n) = (4 + ∛(4n))/2 Yes 7. f(x) = 4/(x - 2) + 2 g(x) = -1/(x + 3) No 8. f(x) = 2/(√x + 1) g(x) = (2/x - 1)^2 Yes Find the inverse of each function. 9. y = √(x - 3) x = √(y - 3) x^2 = y - 3 y = x^2 + 3 10. y = 2x^3 + 3 x = 2y^3 + 3 x - 3 = 2y^3 (x - 3)/2 = y^3 y = ∛((x - 3)/2) 11. x = 2y + 3 x - 3 = 2y y = (x - 3)/2 12. x = 5y + 1 x - 1 = 5y y = (x - 1)/5 7x = 18 x = 7y/18 2x = 7y/9 18x = 7y y = 18x/7 f^(-1)(x) = 18x/7 14. f(x) = x + 3 y = x + 3 x = y + 3 y = x - 3 f^(-1)(x) = x - 3 15. f(x) = 2x - 3 y = 2x - 3 x = 2y - 3 x + 3 = 2y y = (x + 3)/2 f^(-1)(x) = (x + 3)/2 Find the inverse of each function. Then graph the function and its inverse. 17. f(x) = 1/x x = 1/y xy = 1 y = 1/x [Image of graph of f(x) = 1/x and its inverse] 18. f(x) = 1/(x - 1) x = 1/(y - 1) xy - x = 1 xy = 1 + x y = (1 + x)/x [Image of graph of f(x) = 1/(x - 1) and its inverse] 19. f(x) = (x + 2)^2 - 1 x = (y + 2)^2 - 1 x + 1 = (y + 2)^2 ±√(x + 1) = y + 2 y = ±√(x + 1) - 2 [Image of graph of f(x) = (x + 2)^2 - 1 and its inverse] 20. y = 2x - 5 x = 2y - 5 3x = 3y - 5 3x + 5 = y [Image of graph of f(x) = 2x - 5 and its inverse] CLASSWORK Day # Date: OBJECTIVE: Section 5.5-INVERSE FUNCTIONS DEFINITION OF INVERSE FUNCTIONS: Inverse functions "undo" each other... f(x)=x+3 (adds on 3) g(x)=x-3 (takes off 3) f(x)=x^2 (squares x) g(x)=√x (unsquares x) (only works on positive x's!) INVERSE NOTATION: The inverse of f(x) is f^(-1)(x) -1 is not an exponent. Read: "f inverse of x" Graphs of inverse functions... f(x)=x+3 g(x)=x-3 (2,1) (3,0) (0,-3) If we looked at all x values the inverse wouldn't be a function. Graph the inverse: TEST to see if functions are inverses of each other: If g(x) and f(x) are inverses, then g(f(x))=f(g(x))=x If f(x)=x+3 g(x)=x-3 @ f(g(x))=(x-3)+3=x @ g(f(x))=(x+3)-3=x If f(x)=x^2 g(x)=√x @ f(g(x))=√(x)^2=x @ g(f(x))=√(x^2)=x Are these functions inverses? f(x) = 2x - 5 g(x) = (x + 5)/2 f(g(x)) = 2((x + 5)/2) - 5 = x + 5 - 5 = x g(f(x)) = (2x - 5 + 5)/2 = 2x/2 = x Yes! f(x) = x^2 - 3 g(x) = √(x + 3) f(g(x)) = (√(x + 3))^2 - 3 = x + 3 - 3 = x g(f(x)) = √(x^2 - 3 + 3) = √x^2 = x Yes! f(x) = 2x - 5 g(x) = (x + 5)/2 f(g(x)) = 2((x + 5)/2) - 5 = x + 5 - 5 = x g(f(x)) = (2x - 5 + 5)/2 = 2x/2 = x Yes! How do you FIND THE INVERSE OF A FUNCTION? **Step 1:** Switch the x and y around **Step 2:** Solve for y **Check (two options):** 1. Graph each (are they reflections over y=x?) 2. Find f(g(x)) and g(f(x)); do they both equal x? Find the inverse. f(x) = -1/3x + 1 y = -1/3x + 1 x = -1/3y + 1 3(x - 1) = -y y = -3x + 3 f^(-1)(x) = -3x + 3 Check: f(f^(-1)(x)) = -1/3(-3x + 3) + 1 = x - 1 + 1 = x f^(-1)(f(x)) = -3(-1/3x + 1) + 3 = x - 3 + 3 = x y = 2x - 1 x = 2y - 1 x + 1 = 2y y = (x + 1)/2 f^(-1)(x) = (x + 1)/2 h(x) = 2x^5 - 3 y = 2x^5 - 3 x = 2y^5 - 3 (x + 3)/2 = y^5 y = 5√((x + 3)/2)