Math / Algebra

QuestionWhich of the following functions has a graph that is symmetric about the $y$ -axis?
Select all that apply. A. $y=\frac{1}{x}$ B. $y=x^{3}$ C. $y=\sqrt{x}$ D. $y=|x|$

Studdy Solution

STEP 1

1. A function is symmetric about the $y$ -axis if for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph.
2. This symmetry implies that the function satisfies $f(x) = f(-x)$ .

STEP 2

1. Verify symmetry for function A: $y = \frac{1}{x}$ .
2. Verify symmetry for function B: $y = x^3$ .
3. Verify symmetry for function C: $y = \sqrt{x}$ .
4. Verify symmetry for function D: $y = |x|$ .

STEP 3

Verify symmetry for function A: $y = \frac{1}{x}$ .
Calculate $f(-x)$ for $f(x) = \frac{1}{x}$ :
$f(-x) = \frac{1}{-x} = -\frac{1}{x}$
Since $f(x) \neq f(-x)$ , the function $y = \frac{1}{x}$ is not symmetric about the $y$ -axis.

STEP 4

Verify symmetry for function B: $y = x^3$ .
Calculate $f(-x)$ for $f(x) = x^3$ :
$f(-x) = (-x)^3 = -x^3$
Since $f(x) \neq f(-x)$ , the function $y = x^3$ is not symmetric about the $y$ -axis.

STEP 5

Verify symmetry for function C: $y = \sqrt{x}$ .
Calculate $f(-x)$ for $f(x) = \sqrt{x}$ :
The function $\sqrt{x}$ is not defined for negative $x$ , so $f(-x)$ does not exist for negative $x$ .
Since $f(x) \neq f(-x)$ , the function $y = \sqrt{x}$ is not symmetric about the $y$ -axis.

STEP 6

Verify symmetry for function D: $y = |x|$ .
Calculate $f(-x)$ for $f(x) = |x|$ :
$f(-x) = |-x| = |x|$
Since $f(x) = f(-x)$ , the function $y = |x|$ is symmetric about the $y$ -axis.
The function that is symmetric about the $y$ -axis is $\boxed{D}$ .

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QuestionWhich of the following functions has a graph that is symmetric about the $y$ -axis?
Select all that apply. A. $y=\frac{1}{x}$ B. $y=x^{3}$ C. $y=\sqrt{x}$ D. $y=|x|$

Studdy Solution

STEP 1

1. A function is symmetric about the $y$ -axis if for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph.
2. This symmetry implies that the function satisfies $f(x) = f(-x)$ .

STEP 2

1. Verify symmetry for function A: $y = \frac{1}{x}$ .
2. Verify symmetry for function B: $y = x^3$ .
3. Verify symmetry for function C: $y = \sqrt{x}$ .
4. Verify symmetry for function D: $y = |x|$ .

STEP 3

Verify symmetry for function A: $y = \frac{1}{x}$ .
Calculate $f(-x)$ for $f(x) = \frac{1}{x}$ :
$f(-x) = \frac{1}{-x} = -\frac{1}{x}$
Since $f(x) \neq f(-x)$ , the function $y = \frac{1}{x}$ is not symmetric about the $y$ -axis.

STEP 4

Verify symmetry for function B: $y = x^3$ .
Calculate $f(-x)$ for $f(x) = x^3$ :
$f(-x) = (-x)^3 = -x^3$
Since $f(x) \neq f(-x)$ , the function $y = x^3$ is not symmetric about the $y$ -axis.

STEP 5

Verify symmetry for function C: $y = \sqrt{x}$ .
Calculate $f(-x)$ for $f(x) = \sqrt{x}$ :
The function $\sqrt{x}$ is not defined for negative $x$ , so $f(-x)$ does not exist for negative $x$ .
Since $f(x) \neq f(-x)$ , the function $y = \sqrt{x}$ is not symmetric about the $y$ -axis.

STEP 6

Verify symmetry for function D: $y = |x|$ .
Calculate $f(-x)$ for $f(x) = |x|$ :
$f(-x) = |-x| = |x|$
Since $f(x) = f(-x)$ , the function $y = |x|$ is symmetric about the $y$ -axis.
The function that is symmetric about the $y$ -axis is $\boxed{D}$ .

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QuestionWhich of the following functions has a graph that is symmetric about the yyy-axis?Select all that apply. A. y=1xy=\frac{1}{x}y=x1​ B. y=x3y=x^{3}y=x3 C. y=xy=\sqrt{x}y=x​ D. y=∣x∣y=|x|y=∣x∣

Studdy Solution

STEP 1

1. A function is symmetric about the y y y-axis if for every point (x,y) (x, y) (x,y) on the graph, the point (−x,y)(-x, y)(−x,y) is also on the graph.2. This symmetry implies that the function satisfies f(x)=f(−x) f(x) = f(-x) f(x)=f(−x).

STEP 2

1. Verify symmetry for function A: y=1x y = \frac{1}{x} y=x1​.2. Verify symmetry for function B: y=x3 y = x^3 y=x3.3. Verify symmetry for function C: y=x y = \sqrt{x} y=x​.4. Verify symmetry for function D: y=∣x∣ y = |x| y=∣x∣.

STEP 3

STEP 4

STEP 5

STEP 6

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QuestionWhich of the following functions has a graph that is symmetric about the $y$ -axis?
Select all that apply. A. $y=\frac{1}{x}$ B. $y=x^{3}$ C. $y=\sqrt{x}$ D. $y=|x|$

1. A function is symmetric about the $y$ -axis if for every point $(x, y)$ on the graph, the point $(-x, y)$ is also on the graph.
2. This symmetry implies that the function satisfies $f(x) = f(-x)$ .

1. Verify symmetry for function A: $y = \frac{1}{x}$ .
2. Verify symmetry for function B: $y = x^3$ .
3. Verify symmetry for function C: $y = \sqrt{x}$ .
4. Verify symmetry for function D: $y = |x|$ .