Introduction

The roots of a function are the values of the input x for which the function equals zero. For this reason, roots are sometimes also called zeros.

Algebraically, the roots of a function are the solutions for x in the following equation:

f(x)=0

For example, if f(x) = \dfrac{1}{2}x+1, then we can find the roots by solving the equation f(x) = 0, as follows:

\begin{align*} f(x) &= 0 \\[5pt] \dfrac{1}{2}x+1 &= 0 \\[5pt] \dfrac{1}{2}x &= -1 \\[5pt] x &= -2 \end{align*}

So, this function has a single root: x=-2.

Graphically, the roots of a function are the x -intercepts. For example, for the graph of f(x) = \dfrac{1}{2}x+1 shown below, we can see that the function intercepts the x -axis at x=-2.

When shown on a graph, the roots of a function are also called the \boldsymbol x -intercepts.

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Example: Finding the Roots of a Function Given Its Graph

What are the roots of the function shown below?

EXPLANATION

The roots of the function are the x -intercepts. Therefore, the roots are x=-6,-3,0,3,6.

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Practice: Finding the Roots of a Function Given Its Graph

What are the roots of the function shown above?

a
 $1,5$
b
 $-6,8$
c
 $-4,1,5$
d
 $-4,0,5$
e
 $-2,3$
Practice: Finding the Roots of a Function Given Its Graph

Which of the functions shown above has roots $x=0$ and $x=3$ only?

a
 I, II, and III only
b
 II and IV only
c
 I and II only
d
 II and III only
e
 I and III only
Example: Finding the Roots of a Function Given a Table

If f(x) is given by the table shown below, what are its roots?

x -3 -2 -1 0 1 2 3
f(x) 5 0 -2 -4 0 1 3
EXPLANATION

The roots of the function are the values of x such that f(x)=0. In the given table, we see that f(-2)=0 and f(1)=0.

Therefore, the roots of the function are x=-2 and x=1.

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Practice: Finding the Roots of a Function Given a Table

If $f(x)$ is given by the table shown below, what are its roots?

$x$ $-4$ $-3$ $-2$ $0$ $2$ $3$ $4$ $5$
$f(x)$ $16$ $6$ $0$ $1$ $3$ $0$ $16$ $25$
a
 $-4$ and $3$
b
 $-3$ and $5$
c
 $-2$ and $3$
d
 $0$ and $2$
e
 $-3$ and $3$
Practice: Finding the Roots of a Function Given a Table

Which of the following tables describes a function that has $-1$ as a root?

a
$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$
$f(x)$ $-1$ $2$ $3$ $5$ $7$ $4$ $2$
b
$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$
$f(x)$ $6$ $3$ $0$ $2$ $-1$ $3$ $4$
c
$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$
$f(x)$ $2$ $5$ $-1$ $-4$ $3$ $6$ $9$
d
$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$
$f(x)$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$
e
$x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$
$f(x)$ $6$ $3$ $-3$ $-1$ $1$ $2$ $4$
Example: Finding the Roots of a Function Given a Mapping Diagram

What are the roots of the function g(x) defined by the mapping diagram shown below?

EXPLANATION

The roots of the function are the values of x such that g(x)=0. In the given mapping diagram, we see that g(9) = 0.

Therefore, the only root of the function is x=9.

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Practice: Finding the Roots of a Function Given a Mapping Diagram

What are the roots of the function $f(x)$ defined by the mapping diagram shown above?

a
 $-1$ and $4$
b
 $-1,$ $2$ and $4$
c
 $2$ and $4$
d
 $0$ and $2$
e
 $2$ and $3$
Practice: Finding the Roots of a Function Given a Mapping Diagram

What are the roots of the function $f(x)$ defined by the mapping diagram shown above?

a
 $0,$ $2$ and $4$
b
 $0$ and $4$
c
 $2$ and $3$
d
 $f(x)$ has no roots
e
 $1,$ $2$ and $4$
Example: Finding the Roots Given a Function Definition

What is the root of the function f(x)=2-4x?

EXPLANATION

The roots are the solutions to the equation f(x)=0, which we solve as follows:

\begin{align} f(x) &= 0 \\[3pt] 2-4x&=0\\[3pt] 4x&=2\\[3pt] x&=\dfrac12 \end{align}

Therefore, the function has the single root x=\dfrac12.

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Practice: Finding the Roots Given a Function Definition

What is the root of the function $g(x)=\dfrac{x}{2} + 4?$

a
 $4$
b
 $\dfrac{1}{2}$
c
 $-\dfrac{3}{4}$
d
 $-8$
e
 $\dfrac{2}{3}$
Practice: Finding the Roots Given a Function Definition

What is the root of the function $f(x)=5-2x?$

a
 $0$
b
 $\dfrac{5}{2}$
c
 $3$
d
 $-2$
e
 $-\dfrac{2}{5}$
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