Math / Algebra

QuestionDetermine algebraically whether the given function is even, odd, or neither. $f(x)=2 x+|-5 x|$
Choose the correct answer. Even Neither

Studdy Solution

STEP 1

1. A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain.
2. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain.
3. The function given is $f(x) = 2x + |-5x|$ .

STEP 2

1. Determine $f(-x)$ .
2. Compare $f(-x)$ with $f(x)$ to check for evenness.
3. Compare $f(-x)$ with $-f(x)$ to check for oddness.
4. Conclude whether the function is even, odd, or neither.

STEP 3

Determine $f(-x)$ by substituting $-x$ into the function:
$f(-x) = 2(-x) + |-5(-x)|$
Simplify the expression:
$f(-x) = -2x + |5x|$

STEP 4

Compare $f(-x)$ with $f(x)$ to check for evenness:
$f(x) = 2x + |-5x|$
$f(-x) = -2x + |5x|$
For $f(x)$ to be even, $f(-x)$ should equal $f(x)$ . Clearly:
$2x + |-5x| \neq -2x + |5x|$
So, the function is not even.

STEP 5

Compare $f(-x)$ with $-f(x)$ to check for oddness:
$-f(x) = -(2x + |-5x|) = -2x - |-5x|$
$f(-x) = -2x + |5x|$
For $f(x)$ to be odd, $f(-x)$ should equal $-f(x)$ . Clearly:
$-2x + |5x| \neq -2x - |-5x|$
So, the function is not odd.

STEP 6

Since the function is neither even nor odd, we conclude:
The function is $\boxed{\text{neither}}$ .

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QuestionDetermine algebraically whether the given function is even, odd, or neither. $f(x)=2 x+|-5 x|$
Choose the correct answer. Even Neither

Studdy Solution

STEP 1

1. A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain.
2. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain.
3. The function given is $f(x) = 2x + |-5x|$ .

STEP 2

1. Determine $f(-x)$ .
2. Compare $f(-x)$ with $f(x)$ to check for evenness.
3. Compare $f(-x)$ with $-f(x)$ to check for oddness.
4. Conclude whether the function is even, odd, or neither.

STEP 3

Determine $f(-x)$ by substituting $-x$ into the function:
$f(-x) = 2(-x) + |-5(-x)|$
Simplify the expression:
$f(-x) = -2x + |5x|$

STEP 4

Compare $f(-x)$ with $f(x)$ to check for evenness:
$f(x) = 2x + |-5x|$
$f(-x) = -2x + |5x|$
For $f(x)$ to be even, $f(-x)$ should equal $f(x)$ . Clearly:
$2x + |-5x| \neq -2x + |5x|$
So, the function is not even.

STEP 5

Compare $f(-x)$ with $-f(x)$ to check for oddness:
$-f(x) = -(2x + |-5x|) = -2x - |-5x|$
$f(-x) = -2x + |5x|$
For $f(x)$ to be odd, $f(-x)$ should equal $-f(x)$ . Clearly:
$-2x + |5x| \neq -2x - |-5x|$
So, the function is not odd.

STEP 6

Since the function is neither even nor odd, we conclude:
The function is $\boxed{\text{neither}}$ .

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QuestionDetermine algebraically whether the given function is even, odd, or neither. f(x)=2x+∣−5x∣f(x)=2 x+|-5 x|f(x)=2x+∣−5x∣Choose the correct answer. Even Neither

Studdy Solution

STEP 1

STEP 2

1. Determine f(−x) f(-x) f(−x).2. Compare f(−x) f(-x) f(−x) with f(x) f(x) f(x) to check for evenness.3. Compare f(−x) f(-x) f(−x) with −f(x)-f(x)−f(x) to check for oddness.4. Conclude whether the function is even, odd, or neither.

STEP 3

Determine f(−x) f(-x) f(−x) by substituting −x-x−x into the function:f(−x)=2(−x)+∣−5(−x)∣ f(-x) = 2(-x) + |-5(-x)| f(−x)=2(−x)+∣−5(−x)∣Simplify the expression:f(−x)=−2x+∣5x∣ f(-x) = -2x + |5x| f(−x)=−2x+∣5x∣

STEP 4

STEP 5

STEP 6

Since the function is neither even nor odd, we conclude:The function is neither\boxed{\text{neither}}neither​.

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QuestionDetermine algebraically whether the given function is even, odd, or neither. $f(x)=2 x+|-5 x|$
Choose the correct answer. Even Neither

1. Determine $f(-x)$ .
2. Compare $f(-x)$ with $f(x)$ to check for evenness.
3. Compare $f(-x)$ with $-f(x)$ to check for oddness.
4. Conclude whether the function is even, odd, or neither.

Determine $f(-x)$ by substituting $-x$ into the function:
$f(-x) = 2(-x) + |-5(-x)|$
Simplify the expression:
$f(-x) = -2x + |5x|$

Since the function is neither even nor odd, we conclude:
The function is $\boxed{\text{neither}}$ .