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Understanding Domains and Inverses in Functions

Learn to determine domains and ranges, identify one-to-one functions, find inverses algebraically, graph inverse functions, and more. Explore vocabulary like inverse function and function composition.

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Understanding Domains and Inverses in Functions

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  1. DomainsandInverse Functions Sections 1.7 and 1.8

  2. Objectives • Determine the domain and range (where possible) of a function given as an equation. • Determine if a function given as an equation is one-to-one. • Determine if a function given as a graph is one-to-one. • Algebraically find the inverse of a one-to-one function given as an equation.

  3. Objectives • State the domain and range of a function and it inverse. • State the relationships between the domain and range of a function and its inverse • Restrict the domain of a function that is not one-to-one so that an inverse function can be found. • Draw the graph of the inverse function given the graph of the function.

  4. Vocabulary • inverse function • horizontal line test • function composition • one-to-one function

  5. Domain Questions • Does the function have a denominator? • Does the function have a square or even root? • Does the function have a log or ln in it? • Did the function arise from finding an inverse? • Is this a “real world” problem?

  6. Find the domain of the function:

  7. Find the domain of the function:

  8. Find the domain of the function:

  9. Find the domain of the function:

  10. Given the functions and find each of the following:

  11. Determine if the function is one-to-one.

  12. Steps for finding an inverse function. • Change the function notation f(x) to y. • Change all the x’s to y’s and y’s to x’s. • Solve for y. • Replace y with f -1(x).

  13. Find the inverse of the function Find the domains of the function and its inverse.

  14. Find the inverse of the function Find the domains of the function and its inverse.

  15. Find the inverse of the function Find the domains of the function and its inverse.

  16. Find the inverse of the function Find the domains of the function and its inverse.

  17. Draw the graph of the inverse function for the graph of f(x) shown below.

  18. The function is not one-to-one.  Choose the largest possible domain containing the number 100 so that the function restricted to the domain is one-to-one.  Find the inverse function for that restricted function.

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