Questions: Suppose that the function (f) is defined, for all real numbers, as follows. [ f(x)=leftbeginarraycc -5 x+3 text if x leq 2 x-4 text if x>2 endarrayright. ] Graph the function (f). Then determine whether or not the function is continuous. Is the function continuous? Yes No

Suppose that the function (f) is defined, for all real numbers, as follows.
[
f(x)=leftbeginarraycc
-5 x+3 text if x leq 2
x-4 text if x>2
endarrayright.
]

Graph the function (f). Then determine whether or not the function is continuous.

Is the function continuous?
Yes
No

Transcript text: Suppose that the function $f$ is defined, for all real numbers, as follows. \[ f(x)=\left\{\begin{array}{cc} -5 x+3 & \text { if } x \leq 2 \\ x-4 & \text { if } x>2 \end{array}\right. \] Graph the function $f$. Then determine whether or not the function is continuous. Is the function continuous? Yes No

Solution

Solution Steps

Step 1: Define the Piecewise Function

The function $ f(x) $ is defined as: \[ f(x) = \begin{cases} -5x + 3 & \text{if } x \leq 2 \\ x - 4 & \text{if } x > 2 \end{cases} \]

Step 2: Graph the Function for $ x \leq 2 $

For $ x \leq 2 $, the function is $ f(x) = -5x + 3 $.

When $ x = 2 $, $ f(2) = -5(2) + 3 = -10 + 3 = -7 $.
When $ x = 0 $, $ f(0) = -5(0) + 3 = 3 $.

Plot the points (2, -7) and (0, 3) and draw the line segment for $ x \leq 2 $.

Step 3: Graph the Function for $ x > 2 $

For $ x > 2 $, the function is $ f(x) = x - 4 $.

When $ x = 3 $, $ f(3) = 3 - 4 = -1 $.
When $ x = 4 $, $ f(4) = 4 - 4 = 0 $.

Plot the points (3, -1) and (4, 0) and draw the line segment for $ x > 2 $.

Step 4: Check Continuity at $ x = 2 $

To determine if the function is continuous at $ x = 2 $, check the following:

The left-hand limit as $ x $ approaches 2 from the left: $ \lim_{{x \to 2^-}} f(x) = -5(2) + 3 = -7 $.
The right-hand limit as $ x $ approaches 2 from the right: $ \lim_{{x \to 2^+}} f(x) = 2 - 4 = -2 $.
The value of the function at $ x = 2 $: $ f(2) = -7 $.

Since the left-hand limit ($-7$) does not equal the right-hand limit ($-2$), the function is not continuous at $ x = 2 $.