Math / Algebra

QuestionGiven the function below, determine the following. $f(x)=-4 x^{2}+4 x$
Find $f(-x)$ . $f(-x)=$
Select all true statements below. $f(-x)=f(x)$ $f(-x)=-f(x)$ $f$ is an odd function $f$ is an even function $f$ is neither an odd nor even function

Studdy Solution

STEP 1

What is this asking? We're given a parabola equation $f(x)$ and we need to find $f(-x)$ , then figure out if the function is even, odd, or neither! Watch out! Don't mix up even and odd functions!
Remember, even functions are symmetric about the y-axis, like a beautiful butterfly, while odd functions are symmetric about the origin, like a cool spiral.

STEP 2

1. Find $f(-x)$
2. Determine if $f(x)$ is even, odd, or neither

STEP 3

Let's kick things off by substituting $-x$ into our function $f(x)$ .
So, wherever we see an $x$ , we'll replace it with $-x$ .
This is how we find $f(-x)$ !

STEP 4

$f(-x) = -4(-x)^2 + 4(-x)$ Now, let's simplify this!
Remember that $(-x)^2 = x^2$ because a negative times a negative is a positive. $f(-x) = -4(x^2) - 4x$ $f(-x) = -4x^2 - 4x$ So, $f(-x)$ is $-4x^2 - 4x$ . Awesome!

STEP 5

A function is even if $f(-x) = f(x)$ .
Let's check if this is true for our function.
We know $f(x) = -4x^2 + 4x$ and we found that $f(-x) = -4x^2 - 4x$ .
These are clearly not the same, so our function is not even.

STEP 6

A function is odd if $f(-x) = -f(x)$ .
Let's see what $-f(x)$ looks like: $-f(x) = -(-4x^2 + 4x)$ $-f(x) = 4x^2 - 4x$ We found that $f(-x) = -4x^2 - 4x$ , which is not the same as $-f(x)$ .
So, our function is not odd either!

STEP 7

Since $f(-x)$ is not equal to $f(x)$ and $f(-x)$ is not equal to $-f(x)$ , we can confidently say that our function $f(x)$ is neither even nor odd!

STEP 8

$f(-x) = -4x^2 - 4x$ . $f(x)$ is neither an odd nor even function.

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QuestionGiven the function below, determine the following. $f(x)=-4 x^{2}+4 x$
Find $f(-x)$ . $f(-x)=$
Select all true statements below. $f(-x)=f(x)$ $f(-x)=-f(x)$ $f$ is an odd function $f$ is an even function $f$ is neither an odd nor even function

Studdy Solution

STEP 1

STEP 2

1. Find $f(-x)$
2. Determine if $f(x)$ is even, odd, or neither

STEP 3

Let's kick things off by substituting $-x$ into our function $f(x)$ .
So, wherever we see an $x$ , we'll replace it with $-x$ .
This is how we find $f(-x)$ !

STEP 4

$f(-x) = -4(-x)^2 + 4(-x)$ Now, let's simplify this!
Remember that $(-x)^2 = x^2$ because a negative times a negative is a positive. $f(-x) = -4(x^2) - 4x$ $f(-x) = -4x^2 - 4x$ So, $f(-x)$ is $-4x^2 - 4x$ . Awesome!

STEP 5

A function is even if $f(-x) = f(x)$ .
Let's check if this is true for our function.
We know $f(x) = -4x^2 + 4x$ and we found that $f(-x) = -4x^2 - 4x$ .
These are clearly not the same, so our function is not even.

STEP 6

A function is odd if $f(-x) = -f(x)$ .
Let's see what $-f(x)$ looks like: $-f(x) = -(-4x^2 + 4x)$ $-f(x) = 4x^2 - 4x$ We found that $f(-x) = -4x^2 - 4x$ , which is not the same as $-f(x)$ .
So, our function is not odd either!

STEP 7

Since $f(-x)$ is not equal to $f(x)$ and $f(-x)$ is not equal to $-f(x)$ , we can confidently say that our function $f(x)$ is neither even nor odd!

STEP 8

$f(-x) = -4x^2 - 4x$ . $f(x)$ is neither an odd nor even function.

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Studdy Solution

STEP 1

STEP 2

1. Find f(−x)f(-x)f(−x)2. Determine if f(x)f(x)f(x) is even, odd, or neither

STEP 3

Let's **kick things off** by substituting −x-x−x into our function f(x)f(x)f(x).So, wherever we see an xxx, we'll replace it with −x-x−x.This is how we find f(−x)f(-x)f(−x)!

STEP 4

STEP 5

STEP 6

STEP 7

Since f(−x)f(-x)f(−x) is not equal to f(x)f(x)f(x) and f(−x)f(-x)f(−x) is not equal to −f(x)-f(x)−f(x), we can confidently say that our function f(x)f(x)f(x) is **neither even nor odd**!

STEP 8

f(−x)=−4x2−4xf(-x) = -4x^2 - 4xf(−x)=−4x2−4x. f(x)f(x)f(x) is neither an odd nor even function.

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1. Find $f(-x)$
2. Determine if $f(x)$ is even, odd, or neither

Let's kick things off by substituting $-x$ into our function $f(x)$ .
So, wherever we see an $x$ , we'll replace it with $-x$ .
This is how we find $f(-x)$ !

Since $f(-x)$ is not equal to $f(x)$ and $f(-x)$ is not equal to $-f(x)$ , we can confidently say that our function $f(x)$ is neither even nor odd!

$f(-x) = -4x^2 - 4x$ . $f(x)$ is neither an odd nor even function.