3.3: Function composition
- Page ID
- 238563
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Warm-up:
1. Below are three functions: \(f\) and \(g\) are defined by graphs and \(h\) is defined by a table. Use them to answer the following questions.
| \(x\) | -1 | 0 | 1 | 2 | 3 |
| \(h(x)\) | 2 | 1 | 3 | 4 | 1 |
- Estimate \(f \circ g(3)\). Remember, this means \(f(g(3))\).
- Estimate \(g \circ f(3)\).
- Estimate \(g(h(1))\).
- Find \(h(h(-1))\).
- What can you say about \(h(h(h(-1)))\) ?
2. Consider the functions \(f(x)=x^2+x-3 \text { and } g(x)=x-1\).
- Calculate \(f \circ g(4)\).
- Calculate \(f \circ g(-1)\).
- Give an expression of the form \(a x^2+b x+c\) for the function \(f \circ g(x)\). Hint: You need to substitute the \(x\) in the expression for \(f(x)\) with the expression for \(g(x)\), and then simplify.
- Check your answer to (c) by plugging in -1 and 4 for \(x\) and comparing with (a) and (b).
3. Write each of the following functions as a composition of two simpler functions. Be sure to give your simpler functions names, and write down the order in which they should be composed.
- \(h(x)=\frac{1}{x^2-1}\)
- \(v(x)=\sqrt{3 x+10}\)
4. Olive-Harvey College Cafeteria is preparing a big order for an event. The population (in thou-
sands) of the order of hot dog is given by the function \(P=f(t)\) in \(t\) years, and the yearly demand (in thousands) for hot dogs as a function of the population is: \(D=g(P)\). Write a sentence which gives the meaning of each the following expressions.
- \(f(12)=2.4\)
- \(g(3.5)=18.4\)
- \(g(f(4))=12.9\)