Introduction

Let's consider the functions g and f with the mapping diagrams shown below. What will the value f(g(2)) be?

The notation f(g(2)) means that we first find the value of g(2) and then put the result into the function f. From the left diagram above, 2 goes to 3, so g(2)=3. Now, the diagram on the right tells us that 3 goes to -5 . Therefore, we get f(g(2)) = f(3) =-5.


Passing one function as an input to another is known as function composition. This has a special notation, (f \circ g)(x), which simply means f(g(x)).

The key is to use the inner function g(x) for the input into the outer function f(x).

Watch out! The composition of functions is not a product of functions. Note the subtle differences in notation here:

  • f(g(x)) = (f\circ g)(x) means "take g(x) and use it as the input into f(x) ".

  • (fg)(x) = f(x)g(x) means "take f(x) and g(x) and multiply them together".

To evaluate a composite function at a specific point, we evaluate the inner function at the specified value, and then we pass the result to the outer function.

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Example: Evaluating a Composite Function

If f(x) = 2x - 1 and g(x) = 5x^2 - 3x , calculate (g \circ f)(-1).

EXPLANATION

The expression (g \circ f)(-1) is equivalent to g(f(-1)).

To evaluate g(f(-1)), we first compute f(-1) {:}

{\color{blue}f(-1)} = 2(-1)-1 = {\color{blue}-3}

Then, we substitute x=-3 into g(x){:}

\begin{align*} g({\color{blue}f(-1)}) &= g({\color{blue}-3}) \\[5pt] &= 5(-3)^2 - 3(-3) \\[5pt] &= 5(9) + 9 \\[5pt] &= 54 \end{align*}

Therefore, (g \circ f)(-1) =54.

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Practice: Evaluating a Composite Function

If $f(x) = 3x - 1$ and $g(x) = x + 3,$ then $(g \circ f)(1) = $

a
b
c
d
e
Practice: Evaluating a Composite Function

If $f(x) = x^2 + 5$ and $g(x) = 2x - 1,$ then $(f \circ g)(3)=$

a
 $30$
b
 $4x^2-4x+6$
c
 $27$
d
 $2x^2 + 9$
e
 $14$
Practice: Evaluating a Composite Function

If $f(x) =5-x $ and $g(x) = 10 - 2^x,$ then $(f \circ g)(3)=$

a
b
c
d
e
Example: Function Composition With Tables

The functions f and g are defined by the tables below. Find the value of (f \circ g)(4).

x -5 -1 1 4 x -1 2 4 8
g(x) 8 4 2 -1 f(x) 5 -1 -5 -13
EXPLANATION

The expression (f \circ g)(4) is equivalent to f(g(4)).

To evaluate f(g(4)), we first find g(4). From the first table, we have

g(4) = -1.

Then, we substitute x=-1 into f(x). From the second table, we have

f(-1) = 5.

Therefore, (f \circ g)(4) = f(g(4))=5.

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Practice: Function Composition With Tables

The functions $f$ and $g$ are defined by the tables below. Find the value of $(f \circ g)(-2).$

$x$ $-2$ $-1$ $0$ $1$ $x$ $-2$ $1$ $3$ $5$
$g(x)$ $3$ $-2$ $5$ $1$ $f(x)$ $0$ $2$ $-4$ $7$
a
 $-4$
b
 $5$
c
 $0$
d
 $2$
e
 $7$
Practice: Function Composition With Tables
$x$ $1$ $2$ $3$ $4$ $5$ $6$
$g(x)$ $7$ $8$ $5$ $6$ $3$ $7$

The function $f$ is defined as $f(x) = x-2,$ while the function $g$ is defined by the table above. Find the value of $(g \circ f)(4).$

a
b
c
d
e
Practice: Function Composition With Tables

The function $f$ is defined as $f(x) = 1-4x,$ while the function $g$ is defined by the table below. Find the value of $(g \circ f)(1).$

$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$
$g(x)$ $5$ $8$ $9$ $-4$ $-6$ $7$
a
 $8$
b
 $5$
c
 $-6$
d
 $9$
e
 $-4$
Practice: Function Composition With Tables

The function $f$ is defined as $f(x) = 2^x-6,$ while the function $g$ is defined by the table below. Find the value of $(g \circ f)(2).$

$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$
$g(x)$ $-2$ $-1$ $0$ $1$ $2$ $3$
a
 $0$
b
 $1$
c
 $-2$
d
 $-1$
e
 $3$
Example: Composing a Function With Itself

If h(t)=2^t -2t, then calculate (h\circ h)(3).

EXPLANATION

The expression (h \circ h)(3) is equivalent to h(h(3)).

To evaluate h(h(3)), we first compute h(3){:}

{\color{blue}h(3)} = 2^{(3)}-2(3) = {\color{blue}2}

Then, we substitute t=2 into h(t){:}

\begin{align*} h({\color{blue}h(3)}) &= h({\color{blue}2}) \\[5pt] &= 2^{(2)}-2(2)\\[5pt] &= 0 \end{align*}

Therefore, (h\circ h)(3)=0.

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Practice: Composing a Function With Itself

If $g(t)=2t+1,$ then $(g\circ g)(3)=$

a
 $7$
b
 $16$
c
 $49$
d
 $15$
e
 $30$
Practice: Composing a Function With Itself

If $f(z)= z^2 - 2z ,$ then $(f\circ f)(1) = $

a
b
c
d
e
Example: Interpreting Function Composition

A company manufactures school backpacks. The function c(q) gives the total cost, in dollars, of making q units. The function q=s(t) gives the number of units manufactured during the first t hours of a production run. Which of the following is the best interpretation of c(s(6)) in this context?

  1. The number of units manufactured during the first 6 hours of production.
  2. The total manufacturing cost during the first 6 hours of production.
  3. The total cost of manufacturing 6 school backpacks.
EXPLANATION

According to the given function definitions, we have the following:

  • The number s(6) gives the number of units manufactured during the first 6 hours of production.

  • The number c(s(6)) gives the total cost, in dollars, of manufacturing s(6) units.

Combining these two statements, we obtain that c(s(6)) can be interpreted as follows:

\qquad The total manufacturing cost during the first 6 hours of production.

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Practice: Interpreting Function Composition

A computer screen saver consists of a circle that expands. The area of the circle, in square inches, is given by the function $A(r),$ where $r$ is the radius of the circle. The function $r=y(t),$ in turn, gives the radius of the circle after $t$ seconds, in inches. Which of the following is the best interpretation of $A(y(3))$ in this context?

a
 The area of the circle when the radius is $3$ inches.
b
 The area of the circle after $3$ seconds.
c
 The radius of the circle after $3$ seconds.
d
 The diameter of the circle after $3$ seconds.
e
 The diameter of the circle when the radius is $3$ inches.
Practice: Interpreting Function Composition

The function $f(d)$ gives the amount of fuel, in gallons, used by a car after driving for $d$ miles. The function $d=g(t)$ gives the distance, in miles, covered by the car in $t$ hours. Which of the following is the best interpretation of $f(g(2))$ in this context?

a
 The distance covered by car in $2$ hours.
b
 The amount of fuel used by the car for driving for $2$ miles.
c
 The distance covered by the car using $2$ gallons of fuel.
d
 The amount of fuel used by the car after driving for $2$ hours.
e
 The time needed for the car to cover $2$ miles.
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