QuestionDetermine if the following function is even, odd, or neither. $t(x)=-|x|$
Answer Even
Odd Neither

Studdy Solution

STEP 1

What is this asking? We need to figure out if the function $t(x) = -|x|$ is even, odd, or neither! Watch out! Don't mix up even and odd functions!
Remember, it's not the same as even and odd numbers.

STEP 2

1. Define even and odd functions
2. Test for even function
3. Test for odd function

STEP 3

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain of $f$ .
This means if we plug in the opposite of $x$ , we get the same original function value back.

STEP 4

A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain of $f$ .
This means if we plug in the opposite of $x$ , we get the opposite of the original function value back.

STEP 5

Let's test if $t(x)$ is even.
We'll substitute $-x$ into our function: $t(-x) = -|-x|$

STEP 6

The absolute value of any number is its distance from zero, which is always non-negative.
So, $|x|$ and $|-x|$ are the same! $|x| = |-x|$ .
Therefore, $t(-x) = -|-x| = -|x|$

STEP 7

Notice that $t(-x) = -|x|$ , which is the same as our original function $t(x) = -|x|$ .
So, $t(-x) = t(x)$ !

STEP 8

Now, let's see if $t(x)$ is odd.
We need to find $-t(x)$ . $-t(x) = -(-|x|) = |x|$

STEP 9

We already found that $t(-x) = -|x|$ .
Is this equal to $-t(x)$ ?
Nope! We found $-t(x) = |x|$ , and $|x|$ is not the same as $-|x|$ unless $x$ is zero.
So, $t(-x)$ is not equal to $-t(x)$ .

STEP 10

Since $t(-x) = t(x)$ , the function $t(x)$ is even!
It's not odd because $t(-x)$ is not equal to $-t(x)$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

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

Parents Influencer program Contact Policy Terms