Inverse functions Problem set B

Find the inverse function of each function.

1. (a) \(f(x)=5 x + 9\)
2. (b) \(g(x)=\frac{7 - 4 x}{3}\)

\(f(x)=3 x - 11\) and \(g(x)=\frac{x + 11}{3}\). Show that g is the inverse of f.

1. (a) \(f(g(x))=x\)
2. (b) \(g(f(x))=x\)

The function \(f(x)=x^{3} - 9 x^{2} + 4\) is not one-to-one for all x. Use your calculator to find the largest domain of the form \(x \geq k\) for which f has an inverse.

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

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

1. (a) Write down \(f(2)\).
2. (b) Write down \(f^{-1}(27)\).
3. (c) Write down the point on the graph of \(y=f^{-1}(x)\) that matches \((2, 9)\).

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

Find the inverse of each function. State the domain of the inverse.

1. (a) \(f(x)=4e^x-3\)
2. (b) \(g(x)=2\ln x+5\), \(x>0\)

The function f is defined by \(f(x)=x^2+10x+31\), for \(x\geq-5\).

1. (a) Write f in the form \((x-h)^2+k\).
2. (b) Find \(f^{-1}(x)\).
3. (c) State the domain and the range of \(f^{-1}\).

A taxi fare is c(d) = 2.5 + 1.5d pounds for a journey of d kilometres.

1. (a) Find the inverse function, and say what it tells you.
2. (b) Use it to find the distance for a fare of £20, in kilometres to 3 significant figures.

A tank holds V = f(t) = 40 + 2.5t litres of water after t minutes. Mia says the inverse is f⁻¹(x) = 0.4x − 40.

1. (a) Work out \(f(f^{-1}(x))\) using this inverse, and say whether it is correct.
2. (b) Write down the correct inverse.

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 \(F(6)\).
2. (b) Find \(F^{-1}(19)\) and say what it means.
3. (c) Write down the point on the graph of \(y=F^{-1}(x)\) that matches \((10, 23)\).

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 \((40, 52)\).
2. (b) What does this point tell you?

\(f(x)=\frac{6}{x + 2}\), \(x \neq -2\).

1. (a) Find \(f^{-1}(x)\).
2. (b) Find \(f^{-1}(8)\).

The function \(f(x)=x^{3} e^{- x}\), \(x \geq 0\), is not one-to-one. Use your calculator to find the largest domain of the form \(x \geq k\) for which f has an inverse.

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

The function f is defined by \(f(x)=3\ln x+0.5\), for \(x>0\).

1. (a) Find \(f^{-1}(x)\).
2. (b) Solve \(f(x)=f^{-1}(x)\). Give each solution to 3 significant figures.