Question

Use the Vertical line test to determine whether the curve is a graph of a function of x.

Short answer

1. The given curve is a function of x
2. The given curve is not a function of x
3. The given curve is a function of x
4. The given curve is not a function of x

Step-by-step solution

Part a. Step 1. Given Information.

Given that, the curve is-

Part a. Step 2. Concept used.

Vertical Test is used forto determine if the given curveis a graphof a functionor not.

Afunctioncan only have one output i.e., y for each unique input i.e., x.

If a vertical line intersects a curve on anxy-plane more than once, it means for one value of xthe curve has more than one value of y,so, the curve does not represent a function.

If all the vertical lines intersect a curve at most once then the curve represents a function.

Part a. Step 3. To find that the curve is the graph of a function of x.

Draw a vertical line on the curve and check whether at any value of x, the line cut more than one value of y or not.

Hence,

The given curve is a function of x because all the vertical lines intersect the curve at most once.

Part b. Step 1. Given Information.

Given that, the curve is-

Part b. Step 2. Concept used.

Vertical Test is used forto determine if the given curveis a graphof a functionor not.

Afunctioncan only have one output i.e., y for each unique input i.e., x.

If a vertical line intersects a curve on anxy-plane more than once, it means for one value of xthe curve has more than one value of y,so, the curve does not represent a function.

If all the vertical lines intersect a curve at most once then the curve represents a function.

Part b. Step 3. To find that the curve is the graph of a function of x.

Draw a vertical line on the curve and check whether at any value of x, the line cut more than one value of y or not.

Hence,

The given curve is not a function of x because one value of x the curve has more than one value of y.

Part c. Step 1. Given Information.

Given that, the curve is-

Part c. Step 2. Concept used.

Vertical Test is used forto determine if the given curveis a graphof a functionor not.

Afunctioncan only have one output i.e., y for each unique input i.e., x.

If a vertical line intersects a curve on anxy-plane more than once, it means for one value of xthe curve has more than one value of y,so, the curve does not represent a function.

If all the vertical lines intersect a curve at most once then the curve represents a function.

Part c. Step 3. To find that the curve is the graph of a function of x.

Draw a vertical line on the curve and check whether at any value of x, the line cut more than one value of y or not.

Hence,

The given curve is a function of x because all the vertical lines intersect the curve at most once.

Part d. Step 1. Given Information.

Given that, the curve is-

Part d. Step 2. Concept used.

Vertical Test is used forto determine if the given curveis a graphof a functionor not.

Afunctioncan only have one output i.e., y for each unique input i.e., x.

If a vertical line intersects a curve on an xy-plane more than once, it means for one value of x the curve has more than one value of y, so, the curve does not represent a function.

If all the vertical lines intersect a curve at most once then the curve represents a function.

Part d. Step 3. To find that the curve is the graph of a function of x.

Draw a vertical line on the curve and check whether at any value of x, the line cut more than one value of y or not.

Hence,

The given curve is not a function of x because for one value of x the curve has more than one value of y