University:
High School
Course:
High School Algebra
Academic year:

2021
Views:

126

Pages:

6
Author:

Felix Choi

Inverse of a Function In your own words, write the meaning of each vocabulary term. inverse functions: opposite functions To Find an Inverse Switch the variable y = 2x + 3 x = 2y + 3 Solve for y' x = 2y + 3 x - 3 = 2y (x - 3)/2 = y That is the inverse Proof that these are inverses of each other when you plug a function into the other (x) and they completely cancel they are inverses Core Concepts Horizontal Line Test The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once. Inverse is a function: Horizontal line crosses once = Pass the inverse is also a function graph passing the horizontal line test Inverse is not a function: Horizontal line crosses twice = Fail the inverse is not a function graph failing the horizontal line test Essential Question: How can you sketch the graph of the inverse of a function? 1. EXPLORATION: Graphing Functions and Their Inverses Work with a partner. Each pair of functions are inverses of each other. Use a graphing calculator to graph f and g in the same viewing window. What do you notice about the graphs? a. f(x) = 4x + 3 g(x) = (x - 3) / 4 f[g(x)] = 4[(x - 3) / 4] + 3 = x - 3 + 3 = x g[f(x)] = (4x + 3 - 3) / 4 = x Proof that these are inverses b. f(x) = √(x + 1) g(x) = x² - 1, x ≥ 0 f[g(x)] = √(x² - 1 + 1) = √(x²) = x (since x ≥ 0) g[f(x)] = (√(x + 1))² - 1 = x + 1 - 1 = x c. f(x) = √(x - 3) g(x) = x² + 3, x ≥ 0 f[g(x)] = √(x² + 3 - 3) = √(x²) = x (since x ≥ 0) g[f(x)] = (√(x - 3))² + 3 = x - 3 + 3 = x d. f(x) = (4x + 4) / (x + 5) g(x) = (4 - 5x) / (x - 4) See next page 2. EXPLORATION: Sketching Graphs of Inverse Functions Use the graph of f to sketch the graph of g, the inverse function of f, on the same set of coordinate axes. a. graph with a line passing through the points (0, 3) and (1, 1) b. graph with a curve passing through the points (0, 1) and (1, 2) c. graph with a curve passing through the points (3, 0) and (4, 1) d. graph with a curve passing through the points (0, 4) and (1, 1) Extra Practice Solve y = f(x) for x. Then find the input(s) when the output is -6. f(x) = 2x - 1 Listing: y = 2x - 1 +1 +1 y + 1 = 2x (y + 1)/2 = x So, g(x) = (x + 1)/2 or Flip: y = 2x - 1 x = 2y - 1 +1 +1 x + 1 = 2y (x + 1)/2 = y g(x) = (x + 1)/2 input = (-6 + 1)/2 = -5/2 f(x) = 1 - x² Listing: y = 1 - x² -1 -1 y - 1 = -x² -1(y - 1) = x² √(-1(y - 1)) = x So, g(x) = √(1 - x) or Flip: y = 1 - x² x = 1 - y² -1 -1 x - 1 = -y² -1(x - 1) = y² √(-1(x - 1)) = y g(x) = √(1 - x) input = √(1 + 6) = √7 f(x) = (x - 1)³ + 2 Listing: y = (x - 1)³ + 2 -2 -2 y - 2 = (x - 1)³ ³√(y - 2) = x - 1 ³√(y - 2) + 1 = x So, g(x) = ³√(x - 2) + 1 or Flip: y = (x - 1)³ + 2 x = (y - 1)³ + 2 -2 -2 x - 2 = (y - 1)³ ³√(x - 2) = y - 1 ³√(x - 2) + 1 = y g(x) = ³√(x - 2) + 1 input = ³√(-6 - 2) + 1 = -2 + 1 = -1