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AI SL · 12.5

Functions

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12.5.3

Inverse functions

12.5.3

Inverse functions

Target: I can say that a function and its inverse are reflections in the line y = x, and that their coordinates swap.

By the end of this section

Key points

What an inverse does

  • The inverse f⁻¹ reverses f: it takes an output of f back to the input.
  • The graph of y = f⁻¹(x) is the reflection of y = f(x) in the line y = x. Every point (a, b) on f gives the point (b, a) on f⁻¹.
  • ln x and eˣ are inverses of each other, so their graphs are reflections in y = x.

Finding and checking an inverse (HL)

  • To find f⁻¹(x): write y = f(x), make x the subject, then write x in place of y.
  • f(f⁻¹(x)) = f⁻¹(f(x)) = x. Use this to check your answer.
  • f⁻¹(x) is not 1/f(x). The −1 is not a power.

One-to-one functions (HL)

  • A function has an inverse only if it is one-to-one: each output comes from one input.
  • Restrict the domain to make a function one-to-one. Use the calculator to find the turning point first.
  • Choose the side of the turning point the question asks for, such as x ≥ k.

Inverse of a non-linear function (HL)

  • The domain of \(f^{-1}\) is the range of f, and the range of \(f^{-1}\) is the domain of f.
  • For \(f(x)=ae^x+b\), the inverse is \(\ln\!\left(\frac{x-b}{a}\right)\). The horizontal asymptote \(y=b\) of f becomes the vertical asymptote \(x=b\) of \(f^{-1}\).
  • For a quadratic with domain \(x\geq k\), complete the square and take the positive square root. For \(x\leq k\) take the negative root.
  • \(f(x)=f^{-1}(x)\) happens where the graphs meet on the line \(y=x\). Solve \(f(x)=x\) with the calculator.

On the calculator

Finding a minimum or maximum

  • Graph the function in a window that shows the turning point.
  • Use the calculator's minimum (or maximum) tool on the graph.
  • Write down the x-coordinate. That is the k in x ≥ k.

Screenshot and video to come

Solving f(x) = x

  • Enter f(x) as Y1 and x as Y2.
  • Use the intersection tool once for each point where the graphs meet.
  • The x-coordinates solve \(f(x)=f^{-1}(x)\).

Screenshot and video to come

Explore: a function and its inverse

Drag the orange point P along the graph of f. Its mirror image P′ lies on the graph of f⁻¹. Look at the coordinates of both before you read the answer.

f(x)
f⁻¹(x)
Domain of f⁻¹

Drag P, or select it (Tab) and use the arrow keys.

Drag P along the graph of f.

  1. Move P. What do you notice about the coordinates of P and P′? Where would P′ be if P were on the line y = x?
  2. Look at the dashed line from P to P′. What is its angle to the line y = x, and where does it cross that line?
  3. Choose eˣ. What is f⁻¹(1)? Which point of f tells you?
  4. Choose x² and tick the box. Move P to the left of the y-axis. What goes wrong with the reflection, and how does x ≥ 0 fix it?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
A function and its inverse

The point \((2, e^2)\) lies on the graph of \(y=e^x\). Give the matching point on the graph of \(y=\ln x\).

−22468−22468xyy = eˣy = ln x
Working
  1. −22468−22468xyy = eˣy = ln x
    The mirror line

    y = ln x is the reflection of y = eˣ in the line y = x.

    ln x is the inverse of eˣ.

  2. −22468−22468xyy = eˣy = ln x
    Swap the coordinates

    \((2, e^2) \to (e^2, 2) = (7.39, 2)\)

    A reflection in y = x swaps x and y.

  3. Check

    \(\ln(7.39)=2\)

    The point sits on y = ln x, as it should.

Video walkthrough coming