QuestionFind the domain of $f(x)$ , evaluate $f(-2)$ , $f(0)$ , and $f(3)$ , graph $f$ , and check its continuity.

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is defined as a piecewise function. . The function $f(x)$ is defined as $x$ for $-3 \leq x \leq -1$ .
3. The function $f(x)$ is defined as $1$ for $-1 < x <1$ .
4. The function $f(x)$ is defined as $-x$ for $1 \leq x \leq3$ .

STEP 2

To determine the domain of $f$ , we need to consider all the values of $x$ for which $f(x)$ is defined.

STEP 3

The function $f(x)$ is defined for $-3 \leq x \leq -1$ , $-1 < x <1$ , and $1 \leq x \leq3$ .

STEP 4

Combine these intervals to find the domain of $f$ .
$Domain = [-3, -1] \cup (-1,1) \cup [1,3]$

STEP 5

The domain of $f$ is $-3 \leq x \leq3$ .

STEP 6

To evaluate $f(-2)$ , $f(0)$ , and $f(3)$ , we need to plug these values into the function $f(x)$ .

STEP 7

For $f(-2)$ , we use the definition of $f(x)$ for $-3 \leq x \leq -1$ .
$f(-2) =2(-2)$

STEP 8

Calculate the value of $f(-2)$ .
$f(-2) =2(-2) = -4$

STEP 9

For $f()$ , we use the definition of $f(x)$ for $- < x <$ .
$f() =$

STEP 10

For $f(3)$ , we use the definition of $f(x)$ for $\leq x \leq3$ .
$f(3) =2 -3$

STEP 11

Calculate the value of $f(3)$ .
$f(3) = -3 = -$

STEP 12

To graph $f(x)$ , we need to graph each piece of the function on its respective interval.

STEP 13

For $-3 \leq x \leq -$ , the function $f(x) =2x$ is a straight line with slope2.

STEP 14

For $- < x <$ , the function $f(x) =$ is a horizontal line at $y =$ .

STEP 15

For $\leq x \leq3$ , the function $f(x) =2 - x$ is a straight line with negative slope.

STEP 16

Combine these graphs to form the graph of $f(x)$ .

STEP 17

To determine if $f$ is continuous on its domain, we need to check if there are any jumps, breaks, or holes in the graph of $f$ .

STEP 18

The function $f(x)$ is continuous on each of its intervals, but there are jumps at $x = -$ and $x =$ .

STEP 19

Therefore, $f$ is not continuous on its domain.
So, the solutions are(a) The domain of $f$ is $-3 \leq x \leq3$ . (b) $f(-) = -4$ , $f() =1$ , and $f(3) = -1$ . (c) The graph of $f$ is a combination of three lines. (d) $f$ is not continuous on its domain.

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