Introduction

We can plot a function f(x) by setting y=f(x).

To demonstrate, let's consider the function

f(x) = x^2-1, where x is any number taken from the set \{-2, -1, 0, 1,2 \}.

First, we create a table containing the input values:

x -2 -1 0 1 2
y

Given a value of x, we can find the corresponding y -value by substituting x into the formula y=x^2-1. For example,

  • if x=-2 then y=(-2)^2-1=3,

  • if x=-1 then y=(-1)^2-1=0,

  • \cdots

Continuing in the same way, we get the following values:

x -2 -1 0 1 2
y 3 0 -1 0 3

Finally, we plot these (x,y) pairs in the xy -plane:

We don't connect the points since f(x) is defined only for values of x contained within the set \{-2,-1,0,1,2\}.

Let's now take a look at an example where x is unrestricted.

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Example: Plotting a Function

What is the graph of y = f(x), where f(x) = 2x - 3?

EXPLANATION

Recall that the graph of y = 2x - 3 is a straight line, and two distinct points can determine any straight line.

So, given any two distinct values of x, we can find the corresponding y -values by substituting x into the formula y = 2x - 3. For example,

  • if x = 0 then y = 2(0) - 3 = -3, which gives the point (0,-3), and

  • if x = 1 then y = 2(1) - 3 = -1, which gives the point (1,-1).

Finally, we plot these (x,y) pairs in the xy -plane and draw a line through them. This gives the following graph.

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Practice: Plotting a Function

Which of the following shows the graph of $y=f(x),$ where $f(x) = 1+2x$ and $x$ is any number taken from the set $\{0,1,2,3\}?$

a
b
c
d
e
Practice: Plotting a Function

Which of the following is the graph of $y=f(x),$ where $f(x) = 1-2x?$

a
b
c
d
e
Evaluating Functions Using Graphs

Consider the function y=f(x), shown below.

We can determine the values of this function at particular points by reading them off its graph.

  • Since the graph passes through the point (-2,3), we know that f(-2) = 3.

  • Since the graph passes through the point (-1,0), we know that f(-1) = 0.

  • Since the graph passes through the point (0,-1), we know that f(0) = -1.

Notice that the point (2,3) is shown with an empty circle. This means that the function is undefined at the point where x=2.

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Example: Evaluating a Function Given Graphically

The function y = f(x) is shown above. Calculate f(-2) + f(1).

EXPLANATION

We can find the necessary function values using the graph.

From the graph, we note the following:

  • When x=-2, we have y=-2. So, f(-2)=-2.

  • When x=1, the function is undefined because the function has an open circle at this point.

Therefore, since f(1) is undefined, the sum f(-2)+f(1) is undefined too.

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Practice: Evaluating a Function Given Graphically

The function $y = f(x)$ is given by the graph above. Calculate $f(-3)+f(-1).$

a
 $-4$
b
 $-1$
c
 Undefined
d
 $0$
e
 $3$
Practice: Evaluating a Function Given Graphically

The function $y = f(x)$ is shown above. Calculate $f(-1) + f(2).$

a
 $5$
b
 $0$
c
 $-2$
d
 $-4$
e
 Undefined
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