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Warm-Up

Warm-Up. Find the domain of the function. Identify any horizontal and vertical asymptotes, and sketch a graph of the function using a table of values. Find your “match”. And compare your graphs If you got different answers, discuss and come up with a concise solution. Rational Functions.

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Warm-Up

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  1. Warm-Up • Find the domain of the function. Identify any horizontal and vertical asymptotes, and sketch a graph of the function using a table of values.

  2. Find your “match” • And compare your graphs • If you got different answers, discuss and come up with a concise solution.

  3. Rational Functions Finding the Inverse of a Rational Function

  4. Standards: • MM4A1. Students will explore rational functions. • Investigate and explain characteristics of rational functions, including domain, range, zeros, points of discontinuity, intervals of increase and decrease, rates of change, local and absolute extrema, symmetry, asymptotes, and end behavior. • Find inverses of rational functions, discussing domain and range, symmetry, and function composition. • Solve rational equations and inequalities analytically, graphically, and by using appropriate technology.

  5. Inverses of Functions • Take 2 minutes to brainstorm and write down everything that you remember about inverses of functions….

  6. Inverses of Functions • The inverse of a function f(x)=y is the function such that f(y)=x. • The inverse of f(x) is denoted f-1(x) or y-1. • The composition of a function and its inverse always equals x. • The graph of the inverse of a function is the graph of the function, reflected over the identity line (y=x).

  7. Finding the Inverse… • Recall that to find the inverse of a function, solve for x and then “switch” x and y. • Ex: • Check that the composition = x.

  8. Graphing Inverse Functions • Now graph the example function and its inverse. Notice the symmetry of the graphs.

  9. Exploration: • Find the Domain, Range, and intercepts. • Find the inverse. • Find the Domain, Range and intercepts of the inverse. • Make a conjecture about how the characteristics of the inverse relate to the characteristics of the original function.

  10. A rational function, f(x), has the following characteristics: • Domain: • Range: • x-intercept: (7, 0) • y-intercept: (0, 6) What are the characteristics of f-1? • Domain: ______________ • Range: _______________ • x-intercept: ___________ • y-intercept: ___________

  11. Rational Inverses • Cross Multiply. • Solve the equation for x. a. Move all “x” terms to one side. All other terms to the other side. b. factor out x (if applicable) 3. Once solved for x, switch x and y to make a new y-1 = equation.

  12. Examples:

  13. Examples:

  14. Examples:

  15. Graphs • Graph each example function and its inverse. Analyze the symmetry.

  16. TOTD • State the range of the inverse of the function. • Find the x and y intercepts of the inverse. • Find f-1(x) • Use composition to check your solution. Homework on the blog!

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