Math / Algebra

QuestionGiven the function below, determine the following. $f(x)=-5 x^{2}$
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 need to plug in $-x$ into a simple quadratic function and see what happens, then figure out if the function is even, odd, or neither! Watch out! Don't forget your parentheses and exponent rules!
A negative times a negative is a positive!

STEP 2

1. Substitute and Simplify
2. Even or Odd?

STEP 3

Let's substitute $-x$ for $x$ in our function $f(x) = -5x^2$ .
So we're figuring out $f(-x)$ , which means everywhere we see an $x$ , we replace it with $(-x)$ .

STEP 4

$f(-x) = -5(-x)^2$ Remember, squaring a negative number makes it positive!
So $(-x)^2 = (-x) \cdot (-x) = x^2$ . $f(-x) = -5(x^2)$ $f(-x) = -5x^2$

STEP 5

A function is even if $f(-x) = f(x)$ .
A function is odd if $f(-x) = -f(x)$ .

STEP 6

We found that $f(-x) = -5x^2$ .
Our original function was $f(x) = -5x^2$ .
Are these the same?
Yes they are!

STEP 7

Since $f(-x)$ and $f(x)$ are the same, the function is even!

STEP 8

$f(-x) = -5x^2$ .
The correct statement is: $f(-x) = f(x)$ , and $f$ is an even function.

Was this helpful?

QuestionGiven the function below, determine the following. $f(x)=-5 x^{2}$
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 need to plug in $-x$ into a simple quadratic function and see what happens, then figure out if the function is even, odd, or neither! Watch out! Don't forget your parentheses and exponent rules!
A negative times a negative is a positive!

STEP 2

1. Substitute and Simplify
2. Even or Odd?

STEP 3

Let's substitute $-x$ for $x$ in our function $f(x) = -5x^2$ .
So we're figuring out $f(-x)$ , which means everywhere we see an $x$ , we replace it with $(-x)$ .

STEP 4

$f(-x) = -5(-x)^2$ Remember, squaring a negative number makes it positive!
So $(-x)^2 = (-x) \cdot (-x) = x^2$ . $f(-x) = -5(x^2)$ $f(-x) = -5x^2$

STEP 5

A function is even if $f(-x) = f(x)$ .
A function is odd if $f(-x) = -f(x)$ .

STEP 6

We found that $f(-x) = -5x^2$ .
Our original function was $f(x) = -5x^2$ .
Are these the same?
Yes they are!

STEP 7

Since $f(-x)$ and $f(x)$ are the same, the function is even!

STEP 8

$f(-x) = -5x^2$ .
The correct statement is: $f(-x) = f(x)$ , and $f$ is an even function.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

What is this asking? We need to plug in −x-x−x into a simple quadratic function and see what happens, then figure out if the function is even, odd, or neither! Watch out! Don't forget your parentheses and exponent rules!A negative times a negative is a positive!

STEP 2

1. Substitute and Simplify2. Even or Odd?

STEP 3

Let's **substitute** −x-x−x for xxx in our function f(x)=−5x2f(x) = -5x^2f(x)=−5x2.So we're figuring out f(−x)f(-x)f(−x), which means *everywhere* we see an xxx, we replace it with (−x)(-x)(−x).

STEP 4

f(−x)=−5(−x)2 f(-x) = -5(-x)^2 f(−x)=−5(−x)2 Remember, squaring a negative number makes it positive!So (−x)2=(−x)⋅(−x)=x2(-x)^2 = (-x) \cdot (-x) = x^2(−x)2=(−x)⋅(−x)=x2. f(−x)=−5(x2) f(-x) = -5(x^2) f(−x)=−5(x2) f(−x)=−5x2 f(-x) = -5x^2 f(−x)=−5x2

STEP 5

A function is **even** if f(−x)=f(x)f(-x) = f(x)f(−x)=f(x).A function is **odd** if f(−x)=−f(x)f(-x) = -f(x)f(−x)=−f(x).

STEP 6

We found that f(−x)=−5x2f(-x) = -5x^2f(−x)=−5x2.Our original function was f(x)=−5x2f(x) = -5x^2f(x)=−5x2.Are these the same?Yes they are!

STEP 7

Since f(−x)f(-x)f(−x) and f(x)f(x)f(x) are the same, the function is **even**!

STEP 8

f(−x)=−5x2f(-x) = -5x^2f(−x)=−5x2.The correct statement is: f(−x)=f(x)f(-x) = f(x)f(−x)=f(x), and fff is an even function.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to plug in $-x$ into a simple quadratic function and see what happens, then figure out if the function is even, odd, or neither! Watch out! Don't forget your parentheses and exponent rules!
A negative times a negative is a positive!

1. Substitute and Simplify
2. Even or Odd?

Let's substitute $-x$ for $x$ in our function $f(x) = -5x^2$ .
So we're figuring out $f(-x)$ , which means everywhere we see an $x$ , we replace it with $(-x)$ .

$f(-x) = -5(-x)^2$ Remember, squaring a negative number makes it positive!
So $(-x)^2 = (-x) \cdot (-x) = x^2$ . $f(-x) = -5(x^2)$ $f(-x) = -5x^2$

A function is even if $f(-x) = f(x)$ .
A function is odd if $f(-x) = -f(x)$ .

We found that $f(-x) = -5x^2$ .
Our original function was $f(x) = -5x^2$ .
Are these the same?
Yes they are!

Since $f(-x)$ and $f(x)$ are the same, the function is even!

$f(-x) = -5x^2$ .
The correct statement is: $f(-x) = f(x)$ , and $f$ is an even function.