Precalculus by Richard Wright

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1-08 Combinations of Functions

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.6.4

Lightning
Figure 1: credit (pixabay/FelixMittermeier)

Imagine your are outside as a thunderstorm approaches. You see a lightning strike and hear the thunder. Because you have taken important classes in school, you know that you see the lightning almost instantaneously because light travels so fast. However, the sound of the thunder is much slower. By timing the interval between the lightning and thunder, you know you can calculate your distance from the storm. You know that the speed of sound in air is a function of the temperature \(v(T) = 331\sqrt{\frac{T}{273}}\), but that temperature must be in Kelvin. Unfortunately, your only know the temperature in degrees Fahrenheit, but you do know that Kelvin are a function of degrees Celsius (K(C) = C + 273) and degrees Celsius are a function of degrees Fahrenheit (\(C(F) = \frac{5}{9}(F - 32)\)).However, it might be nice to have one combined formula instead of three separate formulas. This lesson goes over ways to combine functions.

Combine Functions Using Algebraic Operations

One way to combine functions is using algebraic operations such as addition, subtraction, multiplication, or division. When combining functions this way, the function outputs are combined. Thus, when adding function outputs, just add the entire functions together. The same for the other operations.

Combine Functions with Algebraic Operations

If two functions f(x) and g(x), then

Example 1: Perform Algebraic Operations on Functions

If f(x) = x2 − 1 and g(x) = x + 1, find (a) f + g (b) fg (c)f·g (d) \(\frac{f}{g}\).

Solution
  1. Add the functions.

    f + g = (x2 − 1) + (x + 1)

    = x2 + x

  2. Subtract the functions.

    fg = (x2 − 1) − (x + 1)

    = x2x − 2

  3. Multiply the functions.

    f·g = (x2 − 1)(x + 1)

    = x3 + x2x − 1

  4. Divide the functions. To simplify, factoring might be useful.

    $$ \frac{f}{g} = \frac{x^2 - 1}{x + 1} $$

    $$ = \frac{(x - 1)(x + 1)}{x + 1} $$

    = x − 1

Try It 1

If f(x) = x2 − 1 and g(x) = x3, find (a) f + g (b) f·g.

Answer

(f + g)(x) = x3 + x2 − 1; (f·g)(x) = x5x3

Combine Functions by Composition

There is another way of combining functions other than using algebraic operations. One way is to use substitution. Combining function using substitution is called composition. The new function created by a composition is called a composite function. Compositions are written as

(fg)(x) = f(g(x))

This is read "f composed with g at x equals f of g of x."

Mathematically, the output of the second function, g, becomes the input of the first function, f. In other words, the second, or inner, function is substituted for the variable in the first, or outer, function. For f(g(x)), the function g is substituted for the variable in f. The order is important because usually, (fg)(x) ≠ (gf)(x).

Note that the range of the inside function needs to be within the domain of the outside function. Less formally, the composition has to make sense in terms of inputs and outputs.

Composition of Functions

(fg)(x) = f(g(x))

Function g is substituted for the variable in function f.

The domain of the composite function fg is all x such that x is in the domain of g and g(x) is in the domain of f.

Example 2: Determine whether Composition of Functions is Commutative

If f(x) = x + 1 and g(x) = x2, find f(g(x)) and g(f(x)). Is the composition of the functions is commutative?

Solution

For f(g(x)), start by substituting, g(x) into f(x).

f(x) = x + 1

f(g(x)) = (x2) + 1

= x2 + 1

For g(f(x)), start by substituting, g(x) into f(x).

g(x) = x2

g(f(x)) = (x + 1)2

= x2 + 2x + 1

Because f(g(x)) ≠ g(f(x)) the operation of function composition is not commutative.

Try It 2

If f(x) = x − 1 and g(x) = 3x, find (fg)(x).

Answer

(fg)(x) = 3x − 1

Example 3: Interpret Composite Functions

The function c(s) gives the number of calories burned completing s sit-ups, and s(t) gives the number of sit-ups a person can complete in t minutes. Interpret c(s(3)).

Solution

The inside expression in the composition is s(3). Because the input to the s-function is time, t = 3 represents 3 minutes, and s(3) is the number of sit-ups completed in 3 minutes.

Using s(3) as the input to the function c(s) gives us the number of calories burned during the number of sit-ups that can be completed in 3 minutes, or simply the number of calories burned in 3 minutes by doing sit-ups.

Example 4: Investigate the Order of Function Composition

Let d(t) represent the miles that can be driven in t hours and g(y) gives the gallons of gas used in driving y miles. Which of these expressions is meaningful: d(g(y)) or g(d(t))?

Solution

The function d(t) is a function whose output is the number of miles driven corresponding to the number of hours driven.

number of miles = d(number of hours)

The function g(y) is a function whose output is the number of gallons used corresponding to the number of miles driven.

number of gallons = g(number of miles)

The expression g(y) takes miles as the input and a number of gallons as the output. The function d(t) requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression d(g(y)) is meaningless.

The expression d(t) takes hours as input and a number of miles driven as the output. The function g(y) requires a number of miles as the input. Using d(t)(miles driven) as an input value for g(y), where gallons of gas depends on miles driven, does make sense. The expression g(d(t)) makes sense, and will yield the number of gallons of gas used, g, driving a certain number of miles, d(t), in x hours.

Example 5: Evaluate a Composition of Functions

Given f(t) = t2t and h(x) = 2x, evaluate f(h(1)).

Solution

This is the composition fh. Substitute h into f.

f(h(x)) = (2x)2 − (2x)

= 4x2 − 2x

Now substitute in the 1.

f(h(1)) = 4(1)2 − 2(1)

= 2

Try It 3

Given f(x) = x2x and g(x) = x + 2, evaluate

  1. g(f(2))
  2. g(f(−3))
Answer

4; 14

Find the Domain of a Composite Function

The domain of f(g(x)), the domain both on the domain of f and g. Working from the outside in, the domain of f is the outputs of g(x). Thus only values of x that produce outputs of g(x) that are in the domain of f(x) can be used. So to find the domain of f(g(x)), start by finding the domain of f. Then find the values of x such that g(x) produces the domain of f.

Domain of a Composite Function

The domain of a composite function f(g(x)) is the set of those inputs x in the domain of g for which g(x) is in the domain of f.

Find the Domain of a Composite Function

Given a function composition f(g(x))

  1. Find the domain of f.
  2. Make the range of g = the domain of f.
  3. Find the values of x to produce this range of g.
  4. Finally, find the domain of g.
  5. The domain of f(g(x)) is the combination of steps 3 and 4.

Example 6: Find the Domain of a Composite Function

Find the domain of (fg)(x) where \(f(x) = \frac{5}{x-1}\) and \(g(x) = \frac{4}{3x-2}\).

Solution

Start by finding the domain of the outer function, f. It will be easier to understand if we substitute g(x) into f.

$$ f(g(x)) = \frac{5}{g(x) - 1} $$

This is a rational function, so the denominator cannot equal zero.

g(x) − 1 ≠ 0

g(x) ≠ 1

Domain of f is g(x) ≠ 1.

That means that the range of g cannot equal 1. So set g ≠ 1 and solve for x.

$$ g(x) = \frac{4}{3x-2} ≠ 1 $$

4 ≠ 3x − 2

6 ≠ 3x

2 ≠ x

Now find the domain of \(g(x) = \frac{4}{3x-2}\). Because it is also rational, the denominator cannot equal 0.

3x − 2 ≠ 0

3x ≠ 2

$$ x ≠ \frac{2}{3} $$

The domain of the composite function (fg)(x) is all real number except \(\frac{2}{3}\) and 2. Written in interval notation, it is

$$ \left(-∞, \frac{2}{3}\right) ∪ \left(\frac{2}{3}, 2\right) ∪ \left(2, ∞\right) $$

Try It 4

Find the domain of (fg)(x) where \(f(x) = \frac{1}{x-2}\) and \(g(x) = \sqrt{x+4}\)

Answer

[−4, 0) ∪ (0, ∞)

Decompose a Composite Function

Sometimes, it might be necessary to decompose, or split apart, a complicated function. One way to do that is to write the function as a composition of two simpler functions. There may be more than one way to decompose a function, so choose the decomposition that seems to be the most expedient. Look for two functions g and h such that f(x) = g(h(x)).

Decompose a Composite Function

Identify two functions g and h such that f(x) = g(h(x)).

  1. Identify an inner function. This is h(x).
  2. Replace the inner function with x. This is the g(x).

Example 7: Decompose a Function

Write \(f(x) = \frac{1}{2x^2 + 1}\) as a composition of two functions.

Solution

A composition is when one function is substituted into another function, so look for a function inside the bigger function. Notice the function is similar to a reciprocal function \(f(x) = \frac{1}{x}\), but with the 2x2 + 1 in place of the x. These would be good candidates for the answer.

The outer function is \(g(x) = \frac{1}{x}\) and the inner function is \(h(x) = 2x^2 + 1\).

Try It 5

Write \(f(x) = \sqrt{\frac{1}{x}}\) as the composition of two functions.

Answer

Sample answer: \(g(x) = \sqrt{x}\) and \(h(x) = \frac{1}{x}\)

Lesson Summary

Combine Functions with Algebraic Operations

If two functions f(x) and g(x), then


Composition of Functions

(fg)(x) = f(g(x))

Function g is substituted for the variable in function f.


Domain of a Composite Function

The domain of a composite function f(g(x)) is the set of those inputs x in the domain of g for which g(x) is in the domain of f.


Find the Domain of a Composite Function

Given a function composition f(g(x))

  1. Find the domain of f.
  2. Make the range of g = the domain of f.
  3. Find the values of x to produce this range of g.
  4. Finally, find the domain of g.
  5. The domain of f(g(x)) is the combination of steps 3 and 4.

Decompose a Composite Function

Identify two functions g and h such that f(x) = g(h(x)).

  1. Identify an inner function. This is h(x).
  2. Replace the inner function with x. This is the g(x).

Helpful videos about this lesson.

Practice Exercises

  1. If the order is reversed when composing two functions, can the result ever be the same as the answer in the original order of the composition? If yes, give an example. If no, explain why not.
  2. If f(x) = 2x + 3 and g(x) = x2 + x, find f + g, fg, fg, and \(\frac{f}{g}\).
  3. If f(x) = x + 3 and g(x) = x2 + 6x + 9, find f + g, fg, fg, and \(\frac{f}{g}\).
  4. If f(x) = −x2 and \(g(x) = \sqrt{3x}\), find f + g, fg, fg, and \(\frac{f}{g}\).
  5. Use each pair of functions to find f(g(x)) and g(f(x)). Simplify your answers.

  6. f(x) = 2x2 and \(g(x) = \sqrt{3x}\)
  7. f(x) = x2 − 2x and \(g(x) = \frac{1}{2}x + 1\)
  8. \(f(x) = 2 - \sqrt{x}\) and g(x) = (2 − x)2
  9. Use graphs of parent functions f(x), g(x), and h(x), shown in figure 2, to evaluate the expressions.

    Figure 2: f(x), g(x), and h(x)
  10. g(h(2))
  11. f(g(1))
  12. Use the functions f(x) = x2 + 1 and g(x) = −3x + 2 to evaluate or find the composite function as indicated.

  13. f(g(x))
  14. (gg)(x)
  15. Find functions g(x) and h(x) so the given function can be expressed as f(x) = g(h(x)).

  16. f(x) = (x + 3)2
  17. \(f(x) = \frac{2}{3x^2}\)
  18. \(f(x) = 2\sqrt{5x^2}\)
  19. Problem Solving

  20. The speed of sound in air is a function of the temperature \(v(T) = 331\sqrt{\frac{T}{273}}\), but that temperature must be in Kelvin. Kelvin is a function of degrees Celsius (K(C) = C + 273) and degrees Celsius is a function of degrees Fahrenheit (\(C(F) = \frac{5}{9}(F - 32)\)).
    1. Find the composition T(F) = (KC)(F) to create a formula to convert degrees Fahrenheit to Kelvin.
    2. Find the composition from part a with the speed of sound. In other words, find (vT)(F).
    3. What is the meaning of the function in part b.
    4. Find the speed of sound in air when the temperature is 65°F.
  21. The number of birds, B, in the backyard is a function of the number of eggs, N that they lay, B(N). The number of eggs laid, E, is a function of time, t, E(t). Which of the following would you do in order to find when the number of birds in the backyard is 25?
    1. Evaluate B(E(25)).
    2. Evaluate E(B(25)).
    3. Solve B(E(t)) = 25.
    4. Solve E(B(N)) = 25.
  22. Mixed Review

  23. (1-07) Describe how the following function is transformed from the original parent function: h(x) = −|x + 3| − 4.
  24. (1-07) Write a function for the following graph:
  25. (1-05) Find the zeros of f(x) = (x + 2)2
  26. (1-05) Find the average rate of change on the interval [x, x + h] for k(x) = x3 + 2

Answers

  1. Yes, f(x) = 2x and \(g(x) = \frac{1}{2}x\)
  2. x2 + 3x + 3; −x2 + x + 3; 2x3 + 5x2 + 3x; \(\frac{2x+3}{x^2+x}\)
  3. x2 + 7x + 12; −x2 − 5x − 6; x3 + 9x2 + 27x + 27; \(\frac{1}{x+3}\)
  4. \(\sqrt{3x} - x^2\); \(-x^2 - \sqrt{3x}\); \(-x^2\sqrt{3x}\); \(-\frac{x^2}{\sqrt{3x}}\)
  5. f(g(x)) = 6x; \(g(f(x)) = x\sqrt{6}\)
  6. \(f(g(x)) = \frac{1}{4}x^2 - 1\); \(g(f(x)) = \frac{1}{2}x^2 - x + 1\)
  7. f(g(x)) = x; g(f(x)) = x
  8. 2
  9. 1
  10. 9x2 − 12x + 5
  11. 9x − 4
  12. g(x) = x2; h(x) = x + 3
  13. \(g(x) = \frac{2}{x}\); h(x) = 3x2
  14. \(g(x) = 2\sqrt{x}\); h(x) = 5x2
  15. \(T(F) = \frac{5}{9}F + \frac{2297}{9}\); \((v ∘ T)(F) = 331\sqrt{\frac{5F + 2297}{2457}}\); speed of sound in air based on temperature in degrees Fahrenheit; 341.9 m/s
  16. c
  17. Reflected over the x-axis, shifted left 3 and down 4
  18. \(f(x) = \sqrt{-x} - 2\)
  19. −2
  20. 3x2 + 3xh + h2