Inverse functions

Inverse functions

An inverse function is a function that reverses all the operations of another function. Therefore, an inverse function has all the points of another function, except that the x and y values are reversed.

Lessons

  • 1.

    • What is “inverse”, and what does “inverse” do to a function?
    • Inverse: switch “x” and “y”
    • Inverse: reflect the original function in the line “y = x”

  • 2.
    Graph an inverse
    Given the graph of y=f(x)y = f\left( x \right) as shown,
    Inverse functions
    a)
    Sketch the graph of the inverse y=f1(x)y = {f^{ - 1}}\left( x \right) in 2 ways:
    i) by reflecting f(x)f\left( x \right) in the line y=xy = x
    ii) by switching the x and y coordinates for each point on f(x)f\left( x \right)

    b)
    Is f(x)f\left( x \right) a function?
    Is f1(x){f^{ - 1}}\left( x \right) a function?


  • 3.
    Inverse of a Quadratic Function
    Consider the quadratic function: f(x)=(x+4)2+2f(x) = (x+4)^2 + 2
    a)
    Graph the function f(x)f\left( x \right) and state the domain and range.

    b)
    Graph the inverse f1(x){f^{ - 1}}\left( x \right) and state the domain and range.

    c)
    Is f1(x){f^{ - 1}}\left( x \right) a function?
    If not, describe how to restrict the domain of f(x)f\left( x \right) so that the inverse of f(x)f\left( x \right) can be a function.


  • 4.
    Determine the equation of the inverse.
    Algebraically determine the equation of the inverse f1(x){f^{ - 1}}\left( x \right), given:
    a)
    f(x)=5x+4f\left( x \right) = - 5x + 4

    b)
    f(x)=(7x8)31f\left( x \right) = {\left( {7x - 8} \right)^3} - 1

    c)
    f(x)=3x2+xf\left( x \right) = \frac{{3x}}{{2 + x}}


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