Composite functions Problem set B One question at a time

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

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

   • \(f(g(x))=f(x^{2})\)
   • \(f(g(x))=7 - 2 x^{2}\)

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

   • \(g(f(x))=g(7 - 2 x)\)
   • \(g(f(x))=4 x^{2} - 28 x + 49\)

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;

   • \((f\circ g)(x)=f(3 x + 2)\)
   • \((f\circ g)(x)=9 x^{2} + 12 x - 1\)

2. (b) find \(g(f(3))\).

   • \(f(3)=4\)
   • \(g(4)=14\)

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

1. (a) \(f(x)=\sqrt{x-6}\), \(x\geq 6\), and \(g(x)=5 x + 1\).

   • The range of g must lie inside the domain of f.
   • \(5 x + 1 \geq 6\)
   • \(x \geq 1\)

2. (b) \(f(x)=\ln(x+3)\), \(x>-3\), and \(g(x)=3 x - 9\).

   • The range of g must lie inside the domain of f.
   • \(3 x - 9 > -3\)
   • \(x > 2\)

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

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

   • \(g(3)=5\)
   • \(f(g(3))=f(5)=7\)

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

   • Draw the line y = 2 on the graph of g.
   • a = 2 or 6

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

1. (a) \(h(x)=(2x-9)^5\)

   • The inside is \(2x-9\). The outer step is the fifth power.

2. (b) \(h(x)=\sqrt{6-x}\)

   • The inside is \(6-x\). The outer step is the square root.

3. (c) \(h(x)=\ln(x^2+4)\)

   • The inside is \(x^2+4\). The outer step is the natural logarithm.

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

• \(f(g(x))=5(x^2-4)-2=5x^2-22\)
• \(5x^2-22=53 \;\Rightarrow\; x^2=15\)
• \(x=\pm\sqrt{15}=\pm3.87\)

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.

   • \(g(f(p))=g(0.9 p)\)
   • \(g(f(p))=0.9 p + 4.5\)

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

   • \(f(g(p))=f(p + 4.5)\)
   • \(f(g(p))=0.9 p + 4.05\)

3. (c) Find both totals for a ticket with p = 60. State which order is cheaper for the customer.

   • \(g(f(60))=58.50\)
   • \(f(g(60))=58.05\)

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)).

   • \(A(r(t))=\pi(2 t + 1.5)^2\)
   • \(A=\pi(2t+1.5)^2\)

2. (b) Find the area of the patch after 3 days, to 3 significant figures.

   • \(t=3: r=7.5\)
   • \(A=\pi\times 7.5^2=177\)

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

• g(u) = 3 when u = −3 or 5 (read g).
• f(x) = −3 gives x = 3 or 5. f(x) = 5 gives x = 1 or 7.
• x = 1, 3, 5 or 7

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

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

   • \(f(g(x))=\frac{1}{(3 x + 2)-6}\)
   • \(f(g(x))=\frac{1}{3 x - 4}\)

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

   • f is not defined when its input is 6.
   • \(3 x + 2=6 \Rightarrow x=\frac{4}{3}\)
   • \(x \neq \frac{4}{3}\)

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.

• \(g(3)=9+k\)
• \(f(g(3))=2 k + 15=21\)
• \(k=3\)

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

• \(f(g(x))=3(x^2-1)+2=3x^2-1\)
• \(g(f(x))=(3x+2)^2-1=9x^2+12x+3\)
• \(3x^2-1=9x^2+12x+3 \;\Rightarrow\; 6x^2+12x+4=0\)
• \(x=-1.58\) or \(x=-0.423\)