- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Functions, graphs and inverses practice question 12
Let . Find .
Careful: is not . What does it mean in terms of the inputs and outputs of ?
is the input with . You do not need a formula for : solve .
Isolate the exponential first, , then take the natural logarithm, , of both sides.
An exact expression, or a decimal to three places.
Theory UBC Math 100 §4.6CLP-1 §0.6Key results
Worked solution
Strategy
The value of an inverse at 5 is the solution of f(x) = 5. Isolate the exponential, undo it with the natural logarithm, then divide by the coefficient of x.
Working
f⁻¹(5) = x exactly when f(x) = 5.
Add 1, then divide by 3, so the exponential stands alone.
Take log of both sides: log(eᵘ) = u.
Check by substituting back.
Answer
The answer is log 2 over 2, about 0.347.
Check
f(0) = 2 and f(0.5) = 3e − 1 ≈ 7.15, so the input giving 5 lies between 0 and 0.5, as 0.347 does.
Where this goes wrong
Taking logarithms before the exponential stands alone, or forgetting to divide by 2 at the end. Get e2x by itself first; then log turns the exponent 2x into an ordinary expression to solve.
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