Composite functions Problem set B

Given \(f(x)=7-2x\) and \(g(x)=x^2\), find:

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

Given \(f(x)=x^{2} - 5\) and \(g(x)=3 x + 2\):

1. (a) find \((f\circ g)(x)\), giving your answer in expanded form;
2. (b) find \(g(f(3))\).

In each part, find the largest possible domain of \((f\circ g)(x)\).

1. (a) \(f(x)=\sqrt{x-6}\), \(x\geq 6\), and \(g(x)=5 x + 1\).
2. (b) \(f(x)=\ln(x+3)\), \(x>-3\), and \(g(x)=3 x - 9\).

The graphs of \(y=f(x)\) and \(y=g(x)\) are shown.

1. (a) Find \(f(g(3))\).
2. (b) Solve \(g(a)=2\).

Write each function h as \(f(g(x))\). State f(x) and g(x), where g is the inner function.

1. (a) \(h(x)=(2x-9)^5\)
2. (b) \(h(x)=\sqrt{6-x}\)
3. (c) \(h(x)=\ln(x^2+4)\)

\(f(x)=5x-2\) and \(g(x)=x^2-4\). Solve \(f(g(x))=53\). Give your answers to 3 significant figures.

A concert ticket has a face value of £p. A voucher takes 10% off the price, and a booking fee of £4.50 is added. Let f(p) = 0.9p be the price after the voucher, and g(p) = p + 4.5 the price after the fee is added.

1. (a) Find g(f(p)), the total when the voucher is used first and the fee is added after.
2. (b) Find f(g(p)), the total when the fee is added first and the voucher is applied to that.
3. (c) Find both totals for a ticket with p = 60. State which order is cheaper for the customer.

A patch of algae on a pond is circular. After t days its radius is r(t) = 1.5 + 2t metres. The area of a circle of radius r is A(r) = πr².

1. (a) Write the area as a function of t, in the form A(r(t)).
2. (b) Find the area of the patch after 3 days, to 3 significant figures.

The graphs of \(y=f(x)\) and \(y=g(x)\) are shown. Solve \(g(f(x))=3\).

Given \(f(x)=\frac{1}{x - 6}\) and \(g(x)=3 x + 2\):

1. (a) find \(f(g(x))\);
2. (b) state the largest possible domain of \(f(g(x))\).

Given \(f(x)=2 x - 3\) and \(g(x)=x^2+k\), where k is a constant, and that \((f\circ g)(3)=21\), find the value of k.

\(f(x)=3x+2\) and \(g(x)=x^2-1\). Solve \(f(g(x))=g(f(x))\). Give your answers to 3 significant figures.