Inverse functions

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  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"
  1. Graph an inverse
    Given the graph of y=f(x)y = f\left( x \right) as shown,
    Inverse functions
    1. 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)
    2. Is f(x)f\left( x \right) a function?
      Is f1(x){f^{ - 1}}\left( x \right) a function?
  2. Inverse of a Quadratic Function
    Consider the quadratic function: f(x)=(x+4)2+2f(x) = (x+4)^2 + 2
    1. Graph the function f(x)f\left( x \right) and state the domain and range.
    2. Graph the inverse f1(x){f^{ - 1}}\left( x \right) and state the domain and range.
    3. 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.
  3. Determine the equation of the inverse.
    Algebraically determine the equation of the inverse f1(x){f^{ - 1}}\left( x \right), given:
    1. f(x)=5x+4f\left( x \right) = - 5x + 4
    2. f(x)=(7x8)31f\left( x \right) = {\left( {7x - 8} \right)^3} - 1
    3. f(x)=3x2+xf\left( x \right) = \frac{{3x}}{{2 + x}}
Topic Notes
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.