Composite functions Problem set A

Given \(f(x)=5x-2\) and \(g(x)=x^2\), find:

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

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

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

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

1. (a) \(f(x)=\sqrt{x-2}\), \(x\geq 2\), and \(g(x)=4 x + 1\).
2. (b) \(f(x)=\ln(x+5)\), \(x>-5\), and \(g(x)=2 x - 7\).

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

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

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

1. (a) \(h(x)=(5x+2)^4\)
2. (b) \(h(x)=\sqrt{3x-7}\)
3. (c) \(h(x)=e^{x^2+1}\)

\(f(x)=2x+5\) and \(g(x)=x^2-3\). Solve \(f(g(x))=21\). Give your answers to 3 significant figures.

A phone case costs £p. A shop takes 15% off in a sale, and adds £6 for delivery. Let f(p) = 0.85p be the price after the sale, and g(p) = p + 6 the price after adding delivery.

1. (a) Find g(f(p)), the total when the sale is applied first and delivery is added after.
2. (b) Find f(g(p)), the total when delivery is added first and the sale is applied to that.
3. (c) Find both totals for a case with p = 40. State which order is cheaper for the customer.

An oil spill is circular. After t hours its radius is r(t) = 2 + 3t 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 spill after 4 hours, to 3 significant figures.

The graphs of \(y=f(x)\) and \(y=g(x)\) are shown. Solve \(g(f(x))=2\).

Given \(f(x)=\frac{1}{x - 4}\) and \(g(x)=2 x + 3\):

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

Given \(f(x)=3 x + 1\) and \(g(x)=x^2+k\), where k is a constant, and that \((f\circ g)(2)=25\), find the value of k.

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