4.6 NOTES & HOMEWORK Inverse of a Function
Rational Exponents and Radical Functions
Algebra 2 Advanced
Objectives:
Name: ________________________
Date: _________________________
(1) Identify when the inverse of a function exists and find its equation.
(2) Determine whether two functions are inverses of each other graphically and
algebraically.
(3) Apply inverse functions in the context of a real-life situation.
Example A: Given a function, how do you find its inverse?
Process of finding the inverse (IF an inverse exists):
From the warm up
1. Replace 𝑓(𝑥) 𝑤𝑖𝑡ℎ y.
𝑓(𝑥) = 2𝑥 + 3.
2. Interchange the x and y variables to create
the inverse function.
3. Solve the new equation for y.
4. Replace y with a function name like g(x).
5. Pay attention to the domain of the original
function and its inverse.
6. [Sometimes you will see the inverse function
of 𝑓(𝑥) 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 𝑓 −1 (𝑥).]
Inverses of Nonlinear Functions
***Because Inverses of Nonlinear Functions are NOT always functions, you must restrict the
domain to work around the issue.
Y
Y
Y
Example B: Consider the function 𝑓(𝑥) = 8𝑥 3 + 1. Find its inverse and then determine whether the
inverse is function.
Example C: Consider the function 𝑓(𝑥) = 2√𝑥 − 3. Determine whether the inverse of f is a function.
Then find the inverse.
Example D: Verify that 𝑓(𝑥) = 3𝑥 − 1 and 𝑔(𝑥) =
𝑥+1
3
are inverse functions.
Some notes/observations:
1. Given a function, f(x), you create its inverse function, g(x), by interchanging the input and output
values of the original function, f(x).
2. This process interchanges the domain and range.
3. If f and g are inverses, then 𝑓(𝑔(𝑥)) = 𝑥 𝑎𝑛𝑑 𝑔(𝑓(𝑥)) = 𝑥.
4. Inverse functions are reflections of each other in the line y = x.
5. To find the inverse function, interchange the x and y variables. This switching of variables creates
the inverse function. Then solve for y.
6. Inverse functions are not a new type of function. Rather, they describe a pair of functions that
undo each other.
4.6 Homework
Algebra 2 Advanced
Name: _______________________________
Date: _______________________________
Find the inverse of the function. Graph both (on the calculator) to verify your solution.
𝑓(𝑥) = 9𝑥 2 , 𝑥 ≤ 0 𝑑𝑜𝑚𝑎𝑖𝑛 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑖𝑜𝑛
1. 𝑓(𝑥) = 6𝑥 − 3
2.
3. 𝑓(𝑥) = (𝑥 − 3)3
4. 𝑓(𝑥) = 2 √𝑥 − 5
5. 𝑓(𝑥) = 2𝑥 2 − 5
3
4𝑥−7
6. 𝑓(𝑥) = −3√
3
Determine whether each pair of functions f and g is inverses. Explain your reasoning.
7.
8.
Use the graph to determine whether the inverse of f is a function. Explain your reasoning.
9.
10.
Determine whether the functions are inverses.
11. 𝑓(𝑥) =
𝑥−3
4
5
, 𝑔(𝑥) = 4𝑥 + 3
𝑥+9
12. 𝑓(𝑥) = √
5
, 𝑔(𝑥) = 5𝑥 5 − 9
Reasoning
13. You and a friend are playing a number-guessing game. You ask your friend to think of a positive
number, square the number multiply the result by 2, and then add 3. Your friend’s final answer is
53. What was the original number chosen? Justify your answer.
Match the function with the graph of its inverse.
3
14. 𝑓(𝑥) = √𝑥 − 4
3
15. 𝑓(𝑥) = √𝑥 + 4
16. 𝑓(𝑥) = √𝑥 + 1 − 3
17. 𝑓(𝑥) = √𝑥 − 1 + 3
Match the graph of the function with the graph of its inverse.
18.
19.
20.
21.
Review Solve.
√4𝑥 − 4 = √5𝑥 − 1 − 1