Comparing Shapes of Functions Quick Check

Question

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Question
Use the table to answer the question.

x f(x)=x3+1 g(x)=x+1
−2 −7 −1
−1 0 0
0 1 1
1 2 2
2 9 3
Compare the functions f(x)=x3+1 and g(x)=x+1 . Which of the following statements is true about the intercepts of the functions?

(1 point)
Responses

Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).
Both functions have a y -intercept of left parenthesis negative 1 comma 0 right parenthesis and an x -intercept of left parenthesis 0 comma 1 right parenthesis .

The graphs intersect at (1,2).
The graphs intersect at left parenthesis 1 comma 2 right parenthesis .

The x- and y-intercepts are different for each function.
The x - and y -intercepts are different for each function.

Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).
Both functions have an x -intercept of left parenthesis negative 1 comma 0 right parenthesis and a y -intercept of left parenthesis 0 comma 1 right parenthesis .
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Answer 1

To compare the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \) based on their intercepts, let's first identify the intercepts for each function:

For \( f(x) = x^3 + 1 \):

1. Y-intercept: This occurs when \( x = 0 \): \[ f(0) = 0^3 + 1 = 1 \implies \text{Y-intercept is } (0, 1). \]

2. X-intercept: This occurs when \( f(x) = 0 \): \[ x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \implies \text{X-intercept is } (-1, 0). \]

For \( g(x) = x + 1 \):

1. Y-intercept: This occurs when \( x = 0 \): \[ g(0) = 0 + 1 = 1 \implies \text{Y-intercept is } (0, 1). \]

2. X-intercept: This occurs when \( g(x) = 0 \): \[ x + 1 = 0 \implies x = -1 \implies \text{X-intercept is } (-1, 0). \]

Summary of intercepts:

• Both functions have a Y-intercept of \( (0, 1) \).
• Both functions have an X-intercept of \( (-1, 0) \).

Now, let's evaluate the options provided:

1. Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).

   • Incorrect.

2. Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).

   • Incorrect.

3. The graphs intersect at (1,2).

   • This is true for \( f(1) \) but not for \( g(1) \), not a statement about intercepts but about points.

4. The graphs intersect at (1,2).

   • Repeated statement, same as above.

5. The x- and y-intercepts are different for each function.

   • Incorrect, they are the same as shown.

6. Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).

   • Correct.

Thus, the correct response is: Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).