6.3.5 Finding the Inverse of a Function Flashcards
Finding the Inverse of a Function
- To determine the inverse of a function algebraically, swap the independent variable (x) and the dependent variable (y) and then solve for y.
- Verify the inverse by composing it with the original function as described in the definition of an inverse.
note
- To find the inverse of a function graphically, you
reflect the curve of the function across the line given
by y = x. - This reflection just swaps the roles of y and x.
- To find the inverse of a function algebraically, first
rename the function y. Then swap y and the
independent variable, which is usually x. - Solve for y to get an expression for the inverse
function. - Since the original function is f ( x ), the inverse is
noted as f^-1( x ). - In order for two functions to be inverses of each
other, all roads must lead to x. - Evaluate the compositions f ( f^-1( x ) ) and f^-1( f ( x ) )
to make sure they both equal x. You must check
both directions.
Which of the following correctly relates the process of algebraically finding the inverse of a function f (x)?
To find the inverse of a function, first rename the function y, then swap y and x, and solve for y to get an expression. If both f −1 ( f (x)) = x and f ( f −1 (x)) = x, this expression is f −1 (x).
Given f (x) = e^x + 2x, which of the following is
f −1 (1)?
0
Given f(x)=2x+3/x−1, find f−1(x).
f−1(x)=x+3/x−2
Given f(x)=1/x+2+5, where x≠−2,find f−1(x).
f−1(x)=1/x−5 −2
To find the inverse of a one-to-one function f (x) graphically, reflect the function’s graph over ____________________.
y=x
Given f (x) = 4x^ 5 + 1, find f −1 (x).
f−1(x)=5√x−1/4
Given f (x) = x ^2, for x > 0, find f −1 (x).
f−1(x)=√x
Given ln(x−1)/3, find f−1(x).
f −1 (x) = e ^3x + 1
Given f (x) = ln (x^ 3 ), find f −1 (x).
f −1 (x) = e^ x / 3
Given f (x) = 2e ^3x + 8, find f −1 (x).
f−1(x)=1/3ln(x−8/2)
