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MAT 153: College Algebra and Trigonometry

Section 2.4 Combining Functions

Often in mathematics, we are interested in using existing functions to create new functions. In this section and the next, we will look at some common ways of building new functions.
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Goals:
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  • F: Be able to determine a composition of functions given in any form (graph, table, equation).
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  • F: Be able to perform arithmetic (sum, difference, product, quotient) on functions given in any form (graph, table, equation).
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  • F: Be able to identify functions and operations that could be combined to produce a given function.
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Example 2.4.1.

You have already seen that function notation uses parentheses to mean something other than multiplication. When examining functions, the function use of parentheses will be the norm. For instance, if \(f(x)=2x\) and \(g(x)=3x^2-2x-5\text{,}\) then we can define \(h(x)=f(x)+g(x)\text{.}\)
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  1. Find \(h(6)=f(6)+g(6)\text{.}\)
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  2. Find an explicit formula for \(h(x)\) (a formula for \(h\) that does not use \(f\) and \(g\)). Use this equation to find \(h(6)\text{.}\) How does this compare to your answer inItem 1?
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Problem 2.4.2.

Refer to \(f\) and \(g\) from Example 2.4.1. Define another new function \(k(x)=2f(x)- 4g(x)\text{.}\) Evaluate \(k(6)\text{.}\) Then find an explicit formula for \(k(x)\) (a formula for \(k\) that does not use \(f\) and \(g\)). Use your formula to evaluate \(k(6)\text{.}\)
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Problem 2.4.3.

Refer to \(f\) and \(g\) from Example 2.4.1. Define \(m(t) = \frac{f(t)}{g(t)}\text{.}\) Evaluate \(m(6)\text{.}\) Then find an explicit formula for \(m(t)\) (a formula for \(m\) that does not use \(f\) and \(g\)).
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Another way to build a new functions is by using the output of one function as the input of another. This is called function composition.
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Definition 2.4.4.

Given functions \(f\) and \(g\text{,}\) define the composition \(p(x)=(f \circ g)(x) = f(g(x))\text{.}\)
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Example 2.4.5.

Refer to \(f\) and \(g\) from Example Example 2.4.1. Define \(p(x)=f(g(x))\text{.}\)
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  1. Find \(p(6)=f(g(6))\text{.}\)
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  2. Find an algebraic formula for \(p(x)\text{.}\) Use this formula to find \(p(6)\text{.}\)
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  3. Define \(q(x)=g(f(x))\text{.}\) Find \(q(6)=g(f(6))\text{.}\) Find an algebraic formula for \(q(x)\text{.}\) Compare these to your answers in parts Item 1 and Item 2. In this case, does it appear that \((f \circ g)(x) = (g \circ f)(x)\text{?}\)
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Problem 2.4.6.

Answer the following questions using the tables for \(a(x)\text{,}\) \(b(x)\) and \(c(x)\) given below.
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Table 2.4.7. Table for \(a(x)\)
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\(x\) \(-2\) \(-1\) 0 1 2 3
\(a(x)\) \(-1\) 1 2 5 \(-4\) \(-1\)
Table 2.4.8. Table for \(b(x)\)
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\(x\) 1 2 5 6
\(b(x)\) 2 3 \(-2\) 1
Table 2.4.9. Table for \(c(x)\)
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\(t\) \(-2\) 0 1 3
\(c(x)\) 5 0 \(-1\) 2
  1. Evaluate \(a(b(1))\text{.}\)
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  2. Evaluate \(b(a(1))\text{.}\)
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  3. Evaluate \(a(b(c(3)))\text{.}\)
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  4. Let \(h(x)=a(x) \cdot b(c(x))\text{.}\) Evaluate \(h(-2)\text{.}\)
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  5. What is \(x\) if \(a(b(x))=5\text{?}\)
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  6. What is \(x\) if \(c(b(x))=2\text{?}\)
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Problem 2.4.10.

For this problem, refer to \(a(t)=4t-7\text{,}\) \(b(t)=-2\cdot 5^t\text{,}\) and \(r(t)\) as given in Table Table 2.4.11.
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Table 2.4.11. \(r(t)\)
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\(t\) 0 1 2 3
\(r(t)\) 25 60 1 -10
  1. Let \(c(x)=\frac{2b(x)-3}{a(x)}\text{.}\) Evaluate \(c(3)\text{.}\) Then find an algebraic formula for \(c(x)\text{.}\)
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  2. Let \(d(x)=x \cdot a(b(x))\text{.}\) Evaluate \(d(2)\text{.}\) Then find an algebraic formula for \(d(x)\text{.}\)
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  3. Let \(s(t)=a(r(t))\text{.}\) Evaluate \(s(3)\text{.}\)
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Problem 2.4.12.

For each function below, determine functions \(f\) and \(g\) so that the function can be written as an operation (sum, difference, product, etc) or composition of \(f\) and \(g\text{.}\)
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  1. \(\displaystyle k(x)=e^{1/x^2}\)
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  2. \(\displaystyle a(x)=\frac{1}{x^2+2}\)
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  3. \(\displaystyle b(x)=\frac{x^2}{10^x}\)
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  4. \(\displaystyle c(x)=x^2(x-2x^2)\)
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  5. \(\displaystyle d(x)=2^x-\frac{1}{2}x^3\)
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  6. \(\displaystyle m(x)=\sqrt[3]{x^4+8}\)
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  7. \(\displaystyle n(x)=x^2e^{x^2}\)
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