Calculus Volume 1 · free preview
Inverse Functions
A one-to-one function can be undone by an inverse function that swaps input and output, reversing the rule and mirroring the graph across the line y equals x.
1. When a function can be undone
When a function can be undone
2. Defining and computing inverses
Defining and computing inverses
3. Graphs of inverse pairs
Graphs of inverse pairs
4. The inverse function theorem
The inverse function theorem
5. The horizontal line test
What does the horizontal line test determine about a function?
6. Computing an inverse
What is the inverse function of f(x) = 3x - 5?
7. Domain restriction
Why is the domain of f(x) = x^2 usually restricted to nonnegative x before speaking of its inverse?
8. Graph of an inverse pair
How is the graph of f inverse related to the graph of a one-to-one function f?
9. Slope of the inverse
The function f is differentiable and one-to-one with f(2) = 8 and f prime of 2 equal to 6. What is (f inverse) prime of 8?
10. Failing the horizontal line test
Which of the following functions fails the horizontal line test on its natural domain and therefore has no inverse there?
11. Inverse of adding seven
The function f(x) = x + 7 undoes itself by subtraction. What is f inverse of 10? Answer with a number.
12. Inverse of cubing
The function f(x) = x^3 has inverse f inverse of x = cube root of x. What number is f inverse of 8? Answer with a number.
13. Swap step result
To invert f(x) = 5x - 3 we swap to x = 5y - 3 and solve. What is y in terms of x? Answer like (x + 3)/5.
14. Composition check
With f(x) = 2x and its inverse g(x) = x/2, what number is g(f(6))? Answer with a number.
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