- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
Functions, graphs and inverses practice question 17
Let . Find without finding a formula for .
A function that only ever increases takes each value once. Is increasing?
is the with . Try small integers.
A whole number.
Theory UBC Math 100 §4.6CLP-1 §0.6Key results
Worked solution
Strategy
A strictly increasing function is one-to-one, so an inverse exists even though solving y = x³ + x for x in general is unpleasant. The value at 10 only needs one solution of a cubic, and it is an integer.
Working
x³ is increasing and x is increasing, so their sum is increasing: if a < b then a³ + a < b³ + b. An increasing function never repeats a value, so f is one-to-one.
Work from the definition of the inverse rather than a formula for it.
x = 2 works, and because f is one-to-one it is the only solution.
Answer
The answer is 2.
Check
f(1) = 2 and f(3) = 30 bracket 10, and since f only increases, the single input giving 10 is 2.
Where this goes wrong
Trying to solve y = x³ + x for x algebraically. The question is designed so that you do not have to: the inverse exists because f is increasing, and the particular value is found by inspection.
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