Inverse functions Problem set A

The graph shows \(y=e^x\), \(y=\ln x\) and the line \(y=x\). The point \(A(1.5, 4.48)\) lies on \(y=e^x\).

1. (a) Write down the coordinates of the point on \(y=\ln x\) that is the reflection of A.
2. (b) State the equation of the mirror line.

The graph of \(y=f(x)\) passes through the points \((1, 3)\), \((2, 5)\), \((4, 9)\) and \((6, 13)\). The function has an inverse. Write down the coordinates of the matching points on the graph of \(y=f^{-1}(x)\).

The graph shows \(y=f(x)\). The point \((3, 8)\) lies on the graph.

1. (a) Write down \(f(3)\).
2. (b) Write down \(f^{-1}(16)\).
3. (c) Write down the point on the graph of \(y=f^{-1}(x)\) that matches \((3, 8)\).

The graph shows \(y=10^x\), \(y=\log_{10}x\) and the line \(y=x\). The point \(P(0.5, 3.16)\) lies on \(y=10^x\).

1. (a) Write down the coordinates of the point on \(y=\log_{10}x\) that is the reflection of P.
2. (b) Use your calculator to find \(\log_{10}3.16\). What does your answer tell you about the point \((3.16, 0.5)\)?

A theatre sells tickets. The cost \(C(n)\) pounds of \(n\) tickets is shown in the table.

1. (a) Write down \(C(3)\).
2. (b) Find \(C^{-1}(50)\) and say what it means.
3. (c) Write down the point on the graph of \(y=C^{-1}(x)\) that matches \((5, 62)\).

The graph shows a straight line \(y=f(x)\) that converts a distance of \(x\) kilometres into \(y\) miles. It passes through \((0, 0)\) and \((50, 31)\).

1. (a) Write down the coordinates of the point on the graph of \(y=f^{-1}(x)\) that matches \((50, 31)\).
2. (b) What does this point tell you?

The graph of \(y=f(x)\), where \(f(x)=0.5x+2\), 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 \(y=x\).
2. (b) Write down the point where \(y=f^{-1}(x)\) meets \(y=x\). Explain your answer.