Inverse functions Problem set B Worked solutions

The graph shows \(y=e^x\), \(y=\ln x\) and the line \(y=x\). The point \(B(3, 1.10)\) lies on \(y=\ln x\).

1. (a) Write down the coordinates of the point on \(y=e^x\) that is the reflection of B.

   • The two graphs are reflections of each other in y = x.
   • Swap the coordinates: (1.10, 3)

2. (b) State the equation of the mirror line.

   • y = x

The graph of \(y=f(x)\) passes through the points \((0, 1)\), \((2, 7)\), \((3, 10)\) and \((5, 16)\). The function has an inverse. Write down the coordinates of the matching points on the graph of \(y=f^{-1}(x)\).

• The graph of \(f^{-1}\) is the reflection of the graph of \(f\) in the line \(y=x\).
• A reflection in \(y=x\) swaps the coordinates: \((a, b)\) becomes \((b, a)\).
• (1, 0), (7, 2), (10, 3), (16, 5)

The graph shows \(y=f(x)\). The point \((2, 9)\) lies on the graph.

1. (a) Write down \(f(2)\).

   • Read up from \(x=2\) to the curve: \(y=9\).

2. (b) Write down \(f^{-1}(27)\).

   • \(f^{-1}(27)\) is the x-value where \(f(x)=27\).
   • Read across from \(y=27\) to the curve: \(x=3\).

3. (c) Write down the point on the graph of \(y=f^{-1}(x)\) that matches \((2, 9)\).

   • Swap the coordinates of \((2, 9)\).
   • (9, 2)

The graph shows \(y=10^x\), \(y=\log_{10}x\) and the line \(y=x\). The point \(Q(5, 0.699)\) lies on \(y=\log_{10}x\).

1. (a) Write down the coordinates of the point on \(y=10^x\) that is the reflection of Q.

   • \(10^x\) and \(\log_{10}x\) are inverses, so their graphs are reflections in \(y=x\).
   • Swap the coordinates: (0.699, 5)

2. (b) Use your calculator to find \(10^{0.699}\). What does your answer tell you about the point \((0.699, 5)\)?

   • From the calculator: \(10^{0.699}=5.00\)
   • So \((0.699, 5)\) lies on \(y=10^x\).

A taxi company charges by the distance travelled. The fare \(F(d)\) pounds for a journey of \(d\) kilometres is shown in the table.

1. (a) Write down \(F(6)\).

   • Read the table at \(d=6\): \(F(6)=15\).

2. (b) Find \(F^{-1}(19)\) and say what it means.

   • \(F^{-1}(19)\) is the distance that costs a fare of £19.
   • In the table, \(F(8)=19\), so \(F^{-1}(19)=8\).
   • A fare of £19 is for a journey of 8 km.

3. (c) Write down the point on the graph of \(y=F^{-1}(x)\) that matches \((10, 23)\).

   • Swap the coordinates of \((10, 23)\).
   • (23, 10)

The graph shows a straight line \(y=g(x)\) that converts an amount of \(x\) pounds into \(y\) dollars. It passes through \((0, 0)\) and \((40, 52)\).

1. (a) Write down the coordinates of the point on the graph of \(y=g^{-1}(x)\) that matches \((40, 52)\).

   • The graph of \(g^{-1}\) is the reflection of the graph of \(g\) in \(y=x\).
   • Swap the coordinates of \((40, 52)\): (52, 40)

2. (b) What does this point tell you?

   • \(g(40)=52\), so \(g^{-1}(52)=40\).
   • 52 dollars is £40. \(g^{-1}\) turns dollars into pounds.

The graph of \(y=f(x)\), where \(f(x)=0.4x+3\), is reflected in the line \(y=x\) to give the graph of \(y=f^{-1}(x)\).

1. (a) Use your calculator to find the point where \(y=f(x)\) meets \(y=x\).

   • Solve \(0.4x+3=x\): \(x=5\).
   • \(y=x=5\), so the point is (5, 5).

2. (b) Write down the point where \(y=f^{-1}(x)\) meets \(y=x\). Explain your answer.

   • Reflecting in \(y=x\) swaps the coordinates.
   • For (5, 5) they are the same, so the point does not move.
   • (5, 5)