AI HL · 12.5
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Composite functions
12.5.3
Each function is a machine. Choose x and follow it through the machines in each order. The function next to x acts first.
On the graph, the orange path goes up to the first function, across to the line y = x so the output becomes the next input, then up or down to the second function.
Move the slider to choose x.
Given \(f(x)=4x+1\) and \(g(x)=x^2\), find (a) \(f(g(x))\) and (b) \(g(f(x))\).
Have a go first. Then open the working one line at a time.
\(f(g(x))=f(x^2)=4 x^{2} + 1\)
The function next to x acts first. Put g(x) into f in place of x.
\(g(f(x))=g(4x+1)=16 x^{2} + 8 x + 1\)
Now put f(x) into g in place of x. Expand the bracket.
\(g(f(1))=25, \quad f(g(1))=5\)
The two orders give different functions. Order matters.
\(f(x)=\ln(x-3)\), \(x>3\), and \(g(x)=3x-4\). Find the largest possible domain of \((f\circ g)(x)\).
Have a go first. Then open the working one line at a time.
The range of g must lie inside the domain of f.
Every output of g becomes an input of f, so g(x) must be greater than 3.
\(g(x)>3 \;\Rightarrow\; 3x-4>3\)
f needs its input to be more than 3.
\(3x>7 \;\Rightarrow\; x>\frac{7}{3}\)
The boundary is \(\frac{7}{3}\), which is 2.33 to 3 significant figures. The domain is x > 2.33.
The graphs of \(y=f(x)\) and \(y=g(x)\) are shown. Solve \(g(f(x))=1\).
Have a go first. Then open the working one line at a time.
\(g(u)=1 \text{ when } u=-3 \text{ or } u=5\)
g(f(x)) = 1 means f(x) is an input where g gives 1. Read g at height 1.
\(f(x)=-3\) \(f(x)=5\)
Now read f at each of those heights.
\(f(x)=-3: x=2, 4\) \(f(x)=5: x=0, 6\)
Each height gives its own x-values.
x = 0, 2, 4 or 6
Four solutions, because each of the two values of u is reached twice by f.
Use the graphs of f and g to find (a) \(f(g(2))\); (b) the values of a for which \(g(a)=3\).
Have a go first. Then open the working one line at a time.
\(g(2)=6\)
Read g at x = 2.
\(f(g(2))=f(6)=-1\)
Read f at x = 6.
a = 1 or 5
Draw the line y = 3 and read where it meets g.
The function h is defined by \(h(x)=(4x-1)^3\). Write h as \(f(g(x))\), where g is the inner function. Find \(g(2)\) and \(h(2)\).
Have a go first. Then open the working one line at a time.
The first thing done to x is \(4x-1\), so \(g(x)=4x-1\).
The inner function is the part you work out first.
What is done to that result is cubing: \(f(x)=x^3\).
The outer function acts on the whole of g(x).
\(f(g(x))=(4x-1)^3=h(x)\)
Forming \(f(g(x))\) gives back h.
\(g(2)=7\) and \(h(2)=f(7)=343\)
Work from the inside out.
\(f(x)=3x-1\) and \(g(x)=x^2+2\). Solve \(f(g(x))=26\). Give the positive solution to 3 significant figures.
Have a go first. Then open the working one line at a time.
\(f(g(x))=3(x^2+2)-1=3x^2+5\)
Put g(x) into f.
\(3x^2+5=26 \;\Rightarrow\; x^2=7\)
Now it is an ordinary equation.
\(x=\pm\sqrt7=\pm2.65\)
A square gives two solutions. The positive one is 2.65.
\(g(2.646)=9\), \(f(9)=26\)
Work through g and then f.