Inverse functions

Lessons

• Introduction
  
  • 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”

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• 1.
  Graph an inverse
  Given the graph of $y = f\left( x \right)$ as shown,
  
  a)

  Sketch the graph of the inverse $y = {f^{ - 1}}\left( x \right)$ in 2 ways:
  i) by reflecting $f\left( x \right)$ in the line $y = x$
  ii) by switching the x and y coordinates for each point on $f\left( x \right)$
  
  
  b)

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

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• 2.
  Inverse of a Quadratic Function
  Consider the quadratic function: $f(x) = (x+4)^2 + 2$
  a)

  Graph the function $f\left( x \right)$ and state the domain and range.
  
  
  b)

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

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

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• 3.
  Determine the equation of the inverse.
  Algebraically determine the equation of the inverse ${f^{ - 1}}\left( x \right)$, given:
  a)

  $f\left( x \right) = - 5x + 4$
  
  
  b)

  $f\left( x \right) = {\left( {7x - 8} \right)^3} - 1$
  
  
  c)

  $f\left( x \right) = \frac{{3x}}{{2 + x}}$
  
  

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