Question

Four Ways to Represent a Function A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function.

To evaluate $f\left(x\right)$, divide the input by 3 and add $\frac{2}{3}$to the result.

Short answer

(a)The algebraic representation is: $f\left(x\right)=\frac{x}{3}+\frac{2}{3}$

(b)The numerical representation is:

$x$	$f\left(x\right)$
2	$\frac{4}{3}$
4	2
6	$\frac{8}{3}$
8	$\frac{10}{3}$

(c)The graphical representation is:

Step-by-step solution

Part a. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

This means that if the object x is in the set of inputs (called the domain) then a function f will map the to exactly one object f(x) in the set of possible outputs (called the range).

Part a. Step 2. Analyze the information given.

We are given that to evaluate $f\left(x\right)$, divide the input by 3 and add $\frac{2}{3}$to the result.

Part a. Step 3. find algebraic representations for the function.

For the given $f\left(x\right)$, the algebraic representation thus can be:$f\left(x\right)=\frac{x}{3}+\frac{2}{3}$

Part b. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$

This means that if the object x is in the set of inputs (called the domain) then a function f will map the to exactly one object f(x) in the set of possible outputs (called the range).

Part b. Step 2. find numerical representations for the function.

From part (a), we have the function, $f\left(x\right)=\frac{x}{3}+\frac{2}{3}$, to write a numerical representation we find few of the values of f for some randomly chosen x- values as shown:

$x$	$f\left(x\right)=\frac{x}{3}+\frac{2}{3}$
2	$\frac{2}{3}+\frac{2}{3}=\frac{4}{3}$
4	$\frac{4}{3}+\frac{2}{3}=2$
6	$\frac{6}{3}+\frac{2}{3}=\frac{8}{3}$
8	$\frac{8}{3}+\frac{2}{3}=\frac{10}{3}$

Part b. Step 3. Evaluate the results.

Thus simplified numerical representation is

$x$	$f\left(x\right)$
2	$\frac{4}{3}$
4	2
6	$\frac{8}{3}$
8	$\frac{10}{3}$

Part c. Step 1. Use the definition of functions.

Function, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

A function is also defined as, given a set of inputs X and a set of possible outputs Y as a set of ordered pairs (x,y). We can write the statement that f is a function from X to Y using the function notationf: $X\to Y$.

This means that if the object x is in the set of inputs (called the domain) then a function f will map the to exactly one object f(x) in the set of possible outputs (called the range).

Part c. Step 2. Analyze the information given.

From part (a), we have the function, $f\left(x\right)=\frac{x}{3}+\frac{2}{3}$, to draw a graphical representation we will use a graphing calculator or tool to get the graph.

Part c. Step 3. find graphical representations for the function.

To draw a graphical representation we will have used a graphing calculator or tool to get the graph as shown: