- Difficulty
- Capstone
- Suggested time
- 8 min
- Marks
- 5
Functions, graphs and inverses practice question 18
A function satisfies the identity below for every . Find .
In the identity, which expression is the input to and which is the output? Which of the two does give back?
is the input to that produces 3. The identity only tells you at inputs of the form , so first find which makes the right-hand side equal 3.
Once you have that , the input to is , not .
A whole number.
Theory UBC Math 100 §4.6CLP-1 §0.6Key results
Worked solution
Strategy
You are never given f directly, only f evaluated at 3x − 1. Find the x for which that output is 3; the corresponding input to f, which is 3x − 1, is the answer.
Working
Find where the known outputs of f equal 3.
Clear the denominator and solve.
At x = 11 the identity says f(32) = 3.
f sends 32 to 3, so f⁻¹ sends 3 back to 32.
Answer
The answer is 32.
Check
Writing u = 3x − 1 gives f(u) = (2u + 17)/(u − 5), and f(32) = 81/27 = 3.
Where this goes wrong
Answering 11. Solving the equation gives the x in the identity, but the argument of f there is 3x − 1 = 32, and that is the value f⁻¹ returns.
Try a similar question
Let . Find without finding a formula for .Saved in this browser only.