Solved on Feb 05, 2024

Determine if the function $f(x) = |x| + 2$ is odd, even, or neither.

STEP 1

Assumptions
1. A function $f(x)$ is even if $f(x) = f(-x)$ for all $x$ in the domain of $f$ .
2. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .
3. The function given is $f(x) = |x| + 2$ .

STEP 2

To determine if the function is even, we need to compare $f(x)$ with $f(-x)$ .

STEP 3

Calculate $f(-x)$ by replacing $x$ with $-x$ in the function.
$f(-x) = |-x| + 2$

STEP 4

Simplify $f(-x)$ using the property of absolute value, which states that $|x| = |-x|$ for all $x$ .
$f(-x) = |x| + 2$

STEP 5

Now, compare $f(-x)$ with $f(x)$ .
Since $f(x) = |x| + 2$ and $f(-x) = |x| + 2$ , we see that $f(x) = f(-x)$ .

STEP 6

Since $f(x) = f(-x)$ for all $x$ , the function $f(x) = |x| + 2$ is an even function.

STEP 7

To determine if the function is odd, we need to check if $f(-x) = -f(x)$ for all $x$ .

STEP 8

Calculate $-f(x)$ by multiplying the function by $-1$ .
$-f(x) = -(|x| + 2)$

STEP 9

Simplify $-f(x)$ .
$-f(x) = -|x| - 2$

STEP 10

Now, compare $-f(x)$ with $f(-x)$ .
Since $f(-x) = |x| + 2$ and $-f(x) = -|x| - 2$ , we see that $f(-x) \neq -f(x)$ .

STEP 11

Since $f(-x) \neq -f(x)$ for all $x$ , the function $f(x) = |x| + 2$ is not an odd function.

STEP 12

Since the function $f(x) = |x| + 2$ is even and not odd, we conclude that the function is even.
The function $f(x) = |x| + 2$ is even.

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Determine if the function $f(x) = |x| + 2$ is odd, even, or neither.

STEP 1

Assumptions
1. A function $f(x)$ is even if $f(x) = f(-x)$ for all $x$ in the domain of $f$ .
2. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .
3. The function given is $f(x) = |x| + 2$ .

STEP 2

To determine if the function is even, we need to compare $f(x)$ with $f(-x)$ .

STEP 3

Calculate $f(-x)$ by replacing $x$ with $-x$ in the function.
$f(-x) = |-x| + 2$

STEP 4

Simplify $f(-x)$ using the property of absolute value, which states that $|x| = |-x|$ for all $x$ .
$f(-x) = |x| + 2$

STEP 5

Now, compare $f(-x)$ with $f(x)$ .
Since $f(x) = |x| + 2$ and $f(-x) = |x| + 2$ , we see that $f(x) = f(-x)$ .

STEP 6

Since $f(x) = f(-x)$ for all $x$ , the function $f(x) = |x| + 2$ is an even function.

STEP 7

To determine if the function is odd, we need to check if $f(-x) = -f(x)$ for all $x$ .

STEP 8

Calculate $-f(x)$ by multiplying the function by $-1$ .
$-f(x) = -(|x| + 2)$

STEP 9

Simplify $-f(x)$ .
$-f(x) = -|x| - 2$

STEP 10

Now, compare $-f(x)$ with $f(-x)$ .
Since $f(-x) = |x| + 2$ and $-f(x) = -|x| - 2$ , we see that $f(-x) \neq -f(x)$ .

STEP 11

Since $f(-x) \neq -f(x)$ for all $x$ , the function $f(x) = |x| + 2$ is not an odd function.

STEP 12

Since the function $f(x) = |x| + 2$ is even and not odd, we conclude that the function is even.
The function $f(x) = |x| + 2$ is even.

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Determine if the function f(x)=∣x∣+2f(x) = |x| + 2f(x)=∣x∣+2 is odd, even, or neither.

STEP 1

STEP 2

To determine if the function is even, we need to compare f(x)f(x)f(x) with f(−x)f(-x)f(−x).

STEP 3

Calculate f(−x)f(-x)f(−x) by replacing xxx with −x-x−x in the function.f(−x)=∣−x∣+2f(-x) = |-x| + 2f(−x)=∣−x∣+2

STEP 4

Simplify f(−x)f(-x)f(−x) using the property of absolute value, which states that ∣x∣=∣−x∣|x| = |-x|∣x∣=∣−x∣ for all xxx.f(−x)=∣x∣+2f(-x) = |x| + 2f(−x)=∣x∣+2

STEP 5

Now, compare f(−x)f(-x)f(−x) with f(x)f(x)f(x).Since f(x)=∣x∣+2f(x) = |x| + 2f(x)=∣x∣+2 and f(−x)=∣x∣+2f(-x) = |x| + 2f(−x)=∣x∣+2, we see that f(x)=f(−x)f(x) = f(-x)f(x)=f(−x).

STEP 6

Since f(x)=f(−x)f(x) = f(-x)f(x)=f(−x) for all xxx, the function f(x)=∣x∣+2f(x) = |x| + 2f(x)=∣x∣+2 is an even function.

STEP 7

To determine if the function is odd, we need to check if f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x) for all xxx.

STEP 8

Calculate −f(x)-f(x)−f(x) by multiplying the function by −1-1−1.−f(x)=−(∣x∣+2)-f(x) = -(|x| + 2)−f(x)=−(∣x∣+2)

STEP 9

Simplify −f(x)-f(x)−f(x).−f(x)=−∣x∣−2-f(x) = -|x| - 2−f(x)=−∣x∣−2

STEP 10

Now, compare −f(x)-f(x)−f(x) with f(−x)f(-x)f(−x).Since f(−x)=∣x∣+2f(-x) = |x| + 2f(−x)=∣x∣+2 and −f(x)=−∣x∣−2-f(x) = -|x| - 2−f(x)=−∣x∣−2, we see that f(−x)≠−f(x)f(-x) \neq -f(x)f(−x)=−f(x).

STEP 11

Since f(−x)≠−f(x)f(-x) \neq -f(x)f(−x)=−f(x) for all xxx, the function f(x)=∣x∣+2f(x) = |x| + 2f(x)=∣x∣+2 is not an odd function.

STEP 12

Since the function f(x)=∣x∣+2f(x) = |x| + 2f(x)=∣x∣+2 is even and not odd, we conclude that the function is even.The function f(x)=∣x∣+2f(x) = |x| + 2f(x)=∣x∣+2 is even.

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Determine if the function $f(x) = |x| + 2$ is odd, even, or neither.

To determine if the function is even, we need to compare $f(x)$ with $f(-x)$ .

Calculate $f(-x)$ by replacing $x$ with $-x$ in the function.
$f(-x) = |-x| + 2$

Simplify $f(-x)$ using the property of absolute value, which states that $|x| = |-x|$ for all $x$ .
$f(-x) = |x| + 2$

Now, compare $f(-x)$ with $f(x)$ .
Since $f(x) = |x| + 2$ and $f(-x) = |x| + 2$ , we see that $f(x) = f(-x)$ .

Since $f(x) = f(-x)$ for all $x$ , the function $f(x) = |x| + 2$ is an even function.

To determine if the function is odd, we need to check if $f(-x) = -f(x)$ for all $x$ .

Calculate $-f(x)$ by multiplying the function by $-1$ .
$-f(x) = -(|x| + 2)$

Simplify $-f(x)$ .
$-f(x) = -|x| - 2$

Now, compare $-f(x)$ with $f(-x)$ .
Since $f(-x) = |x| + 2$ and $-f(x) = -|x| - 2$ , we see that $f(-x) \neq -f(x)$ .

Since $f(-x) \neq -f(x)$ for all $x$ , the function $f(x) = |x| + 2$ is not an odd function.

Since the function $f(x) = |x| + 2$ is even and not odd, we conclude that the function is even.
The function $f(x) = |x| + 2$ is even.