Composite functions Problem set A One question at a time

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

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

   • \(f(g(x))=f(x^{2})\)
   • \(f(g(x))=5 x^{2} - 2\)

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

   • \(g(f(x))=g(5 x - 2)\)
   • \(g(f(x))=25 x^{2} - 20 x + 4\)

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;

   • \((f\circ g)(x)=f(2 x - 1)\)
   • \((f\circ g)(x)=4 x^{2} - 4 x + 4\)

2. (b) find \(g(f(2))\).

   • \(f(2)=7\)
   • \(g(7)=13\)

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

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

   • The range of g must lie inside the domain of f.
   • \(4 x + 1 \geq 2\)
   • \(x \geq \frac{1}{4}\)

2. (b) \(f(x)=\ln(x+5)\), \(x>-5\), and \(g(x)=2 x - 7\).

   • The range of g must lie inside the domain of f.
   • \(2 x - 7 > -5\)
   • \(x > 1\)

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

1. (a) Find \(f(g(3))\).

   • \(g(3)=4\)
   • \(f(g(3))=f(4)=-3\)

2. (b) Solve \(g(a)=1\).

   • Draw the line y = 1 on the graph of g.
   • a = 0 or 4

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

1. (a) \(h(x)=(5x+2)^4\)

   • The inside is \(5x+2\). The outer step is the fourth power.

2. (b) \(h(x)=\sqrt{3x-7}\)

   • The inside is \(3x-7\). The outer step is the square root.

3. (c) \(h(x)=e^{x^2+1}\)

   • The inside is \(x^2+1\). The outer step is the exponential.

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

• \(f(g(x))=2(x^2-3)+5=2x^2-1\)
• \(2x^2-1=21 \;\Rightarrow\; x^2=11\)
• \(x=\pm\sqrt{11}=\pm3.32\)

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.

   • \(g(f(p))=g(0.85 p)\)
   • \(g(f(p))=0.85 p + 6\)

2. (b) Find f(g(p)), the total when delivery is added first and the sale is applied to that.

   • \(f(g(p))=f(p + 6)\)
   • \(f(g(p))=0.85 p + 5.1\)

3. (c) Find both totals for a case with p = 40. State which order is cheaper for the customer.

   • \(g(f(40))=40.00\)
   • \(f(g(40))=39.10\)

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)).

   • \(A(r(t))=\pi(3 t + 2)^2\)
   • \(A=\pi(3t+2)^2\)

2. (b) Find the area of the spill after 4 hours, to 3 significant figures.

   • \(t=4: r=14\)
   • \(A=\pi\times 14^2=616\)

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

• g(u) = 2 when u = −1 or 3 (read g).
• f(x) = −1 gives x = 2. f(x) = 3 gives x = 0 or 4.
• x = 0, 2 or 4

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

1. (a) find \(f(g(x))\);

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

2. (b) state the largest possible domain of \(f(g(x))\).

   • f is not defined when its input is 4.
   • \(2 x + 3=4 \Rightarrow x=\frac{1}{2}\)
   • \(x \neq \frac{1}{2}\)

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.

• \(g(2)=4+k\)
• \(f(g(2))=3 k + 13=25\)
• \(k=4\)

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

• \(f(g(x))=2(x^2+1)-3=2x^2-1\)
• \(g(f(x))=(2x-3)^2+1=4x^2-12x+10\)
• \(2x^2-1=4x^2-12x+10 \;\Rightarrow\; 2x^2-12x+11=0\)
• \(x=1.13\) or \(x=4.87\)