QuestionIdentify which functions are symmetric about the $y$ -axis: A. $y=x^{3}$ , B. $y=\sqrt{x}$ , C. $y=\frac{1}{x}$ , D. $y=|x|$ .

Studdy Solution

STEP 1

Assumptions1. A function is symmetric about the y-axis if and only if f(x) = f(-x) for all x in the domain of f. . We have four functions to test for symmetry about the y-axis $y=x^{3}$ , $y=\sqrt{x}$ , $y=\frac{1}{x}$ , and $y=|x|$ .

STEP 2

We will test each function by replacing $x$ with $-x$ and see if we get the same function.
First, let's test the function $y=x^{}$ .
$f(-x) = (-x)^{} = -x^{}$

STEP 3

We can see that $f(-x) \neq f(x)$ for the function $y=x^{3}$ , so this function is not symmetric about the y-axis.

STEP 4

Next, let's test the function $y=\sqrt{x}$ .
Since the square root of a negative number is not a real number, $f(-x)$ is not defined for $y=\sqrt{x}$ . Therefore, this function is not symmetric about the y-axis.

STEP 5

Now, let's test the function $y=\frac{1}{x}$ .
$f(-x) = \frac{1}{-x} = -\frac{1}{x}$

STEP 6

We can see that $f(-x) \neq f(x)$ for the function $y=\frac{1}{x}$ , so this function is not symmetric about the y-axis.

STEP 7

Finally, let's test the function $y=|x|$ .
$f(-x) = |-x| = |x|$

STEP 8

We can see that $f(-x) = f(x)$ for the function $y=|x|$ , so this function is symmetric about the y-axis.
Therefore, the only function that is symmetric about the y-axis is $y=|x|$ .

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QuestionIdentify which functions are symmetric about the $y$ -axis: A. $y=x^{3}$ , B. $y=\sqrt{x}$ , C. $y=\frac{1}{x}$ , D. $y=|x|$ .

Studdy Solution

STEP 1

Assumptions1. A function is symmetric about the y-axis if and only if f(x) = f(-x) for all x in the domain of f. . We have four functions to test for symmetry about the y-axis $y=x^{3}$ , $y=\sqrt{x}$ , $y=\frac{1}{x}$ , and $y=|x|$ .

STEP 2

We will test each function by replacing $x$ with $-x$ and see if we get the same function.
First, let's test the function $y=x^{}$ .
$f(-x) = (-x)^{} = -x^{}$

STEP 3

We can see that $f(-x) \neq f(x)$ for the function $y=x^{3}$ , so this function is not symmetric about the y-axis.

STEP 4

Next, let's test the function $y=\sqrt{x}$ .
Since the square root of a negative number is not a real number, $f(-x)$ is not defined for $y=\sqrt{x}$ . Therefore, this function is not symmetric about the y-axis.

STEP 5

Now, let's test the function $y=\frac{1}{x}$ .
$f(-x) = \frac{1}{-x} = -\frac{1}{x}$

STEP 6

We can see that $f(-x) \neq f(x)$ for the function $y=\frac{1}{x}$ , so this function is not symmetric about the y-axis.

STEP 7

Finally, let's test the function $y=|x|$ .
$f(-x) = |-x| = |x|$

STEP 8

We can see that $f(-x) = f(x)$ for the function $y=|x|$ , so this function is symmetric about the y-axis.
Therefore, the only function that is symmetric about the y-axis is $y=|x|$ .

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QuestionIdentify which functions are symmetric about the yyy-axis: A. y=x3y=x^{3}y=x3, B. y=xy=\sqrt{x}y=x​, C. y=1xy=\frac{1}{x}y=x1​, D. y=∣x∣y=|x|y=∣x∣.

Studdy Solution

STEP 1

Assumptions1. A function is symmetric about the y-axis if and only if f(x) = f(-x) for all x in the domain of f. . We have four functions to test for symmetry about the y-axis y=x3y=x^{3}y=x3, y=xy=\sqrt{x}y=x​, y=1xy=\frac{1}{x}y=x1​, and y=∣x∣y=|x|y=∣x∣.

STEP 2

We will test each function by replacing xxx with −x-x−x and see if we get the same function.First, let's test the function y=xy=x^{}y=x.f(−x)=(−x)=−xf(-x) = (-x)^{} = -x^{}f(−x)=(−x)=−x

STEP 3

We can see that f(−x)≠f(x)f(-x) \neq f(x)f(−x)=f(x) for the function y=x3y=x^{3}y=x3, so this function is not symmetric about the y-axis.

STEP 4

Next, let's test the function y=xy=\sqrt{x}y=x​.Since the square root of a negative number is not a real number, f(−x)f(-x)f(−x) is not defined for y=xy=\sqrt{x}y=x​. Therefore, this function is not symmetric about the y-axis.

STEP 5

Now, let's test the function y=1xy=\frac{1}{x}y=x1​.f(−x)=1−x=−1xf(-x) = \frac{1}{-x} = -\frac{1}{x}f(−x)=−x1​=−x1​

STEP 6

We can see that f(−x)≠f(x)f(-x) \neq f(x)f(−x)=f(x) for the function y=1xy=\frac{1}{x}y=x1​, so this function is not symmetric about the y-axis.

STEP 7

Finally, let's test the function y=∣x∣y=|x|y=∣x∣.f(−x)=∣−x∣=∣x∣f(-x) = |-x| = |x|f(−x)=∣−x∣=∣x∣

STEP 8

We can see that f(−x)=f(x)f(-x) = f(x)f(−x)=f(x) for the function y=∣x∣y=|x|y=∣x∣, so this function is symmetric about the y-axis.Therefore, the only function that is symmetric about the y-axis is y=∣x∣y=|x|y=∣x∣.

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QuestionIdentify which functions are symmetric about the $y$ -axis: A. $y=x^{3}$ , B. $y=\sqrt{x}$ , C. $y=\frac{1}{x}$ , D. $y=|x|$ .

Assumptions1. A function is symmetric about the y-axis if and only if f(x) = f(-x) for all x in the domain of f. . We have four functions to test for symmetry about the y-axis $y=x^{3}$ , $y=\sqrt{x}$ , $y=\frac{1}{x}$ , and $y=|x|$ .

We will test each function by replacing $x$ with $-x$ and see if we get the same function.
First, let's test the function $y=x^{}$ .
$f(-x) = (-x)^{} = -x^{}$

We can see that $f(-x) \neq f(x)$ for the function $y=x^{3}$ , so this function is not symmetric about the y-axis.

Next, let's test the function $y=\sqrt{x}$ .
Since the square root of a negative number is not a real number, $f(-x)$ is not defined for $y=\sqrt{x}$ . Therefore, this function is not symmetric about the y-axis.

Now, let's test the function $y=\frac{1}{x}$ .
$f(-x) = \frac{1}{-x} = -\frac{1}{x}$

We can see that $f(-x) \neq f(x)$ for the function $y=\frac{1}{x}$ , so this function is not symmetric about the y-axis.

Finally, let's test the function $y=|x|$ .
$f(-x) = |-x| = |x|$

We can see that $f(-x) = f(x)$ for the function $y=|x|$ , so this function is symmetric about the y-axis.
Therefore, the only function that is symmetric about the y-axis is $y=|x|$ .