Question
compare the functions f(x) = x^3 + 1 and g(x) = x + 1. which of the following statements is true about the intercepts of the functions?
a the x- and y- intercepts are different for each function
b the graphs intersect a (1,2)
c both functions have an x-intercept of (-1,0) and a y-intercept of (0,1)
d both functions have a y-intercept of (-1,0) and an x-intercept of (0,1)
a the x- and y- intercepts are different for each function
b the graphs intersect a (1,2)
c both functions have an x-intercept of (-1,0) and a y-intercept of (0,1)
d both functions have a y-intercept of (-1,0) and an x-intercept of (0,1)
Answers
GPT-4o mini
Let's analyze the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \) to determine their intercepts.
### Finding the intercepts:
**1. For \( f(x) = x^3 + 1 \):**
- **Y-intercept**: Set \( x = 0 \):
\[
f(0) = 0^3 + 1 = 1
\]
So, the y-intercept is \( (0, 1) \).
- **X-intercept**: Set \( f(x) = 0 \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1
\]
So, the x-intercept is \( (-1, 0) \).
**2. For \( g(x) = x + 1 \):**
- **Y-intercept**: Set \( x = 0 \):
\[
g(0) = 0 + 1 = 1
\]
So, the y-intercept is \( (0, 1) \).
- **X-intercept**: Set \( g(x) = 0 \):
\[
x + 1 = 0 \implies x = -1
\]
So, the x-intercept is \( (-1, 0) \).
### Summary of intercepts:
- \( f(x) \) has:
- Y-intercept: \( (0, 1) \)
- X-intercept: \( (-1, 0) \)
- \( g(x) \) has:
- Y-intercept: \( (0, 1) \)
- X-intercept: \( (-1, 0) \)
### Analyzing the statements:
a. **The x- and y-intercepts are different for each function.**
- This is **false** because both functions have the same intercepts.
b. **The graphs intersect at (1, 2).**
- We can check by substituting \( x = 1 \) into both functions:
\[
f(1) = 1^3 + 1 = 2 \quad \text{and} \quad g(1) = 1 + 1 = 2
\]
- The graphs do intersect at \( (1, 2) \), so this is **true**.
c. **Both functions have an x-intercept of (-1, 0) and a y-intercept of (0, 1).**
- This is **true** since both functions have these intercepts.
d. **Both functions have a y-intercept of (-1, 0) and an x-intercept of (0, 1).**
- This is **false** because the y-intercept of \( f(x) \) and \( g(x) \) is \( (0, 1) \), not \( (-1, 0) \).
### Conclusion:
- Statements b and c are true, but considering the context, since the question asks which statements are true without specifying, the most explicit intersection result would be:
**b. The graphs intersect at (1, 2).**
### Finding the intercepts:
**1. For \( f(x) = x^3 + 1 \):**
- **Y-intercept**: Set \( x = 0 \):
\[
f(0) = 0^3 + 1 = 1
\]
So, the y-intercept is \( (0, 1) \).
- **X-intercept**: Set \( f(x) = 0 \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1
\]
So, the x-intercept is \( (-1, 0) \).
**2. For \( g(x) = x + 1 \):**
- **Y-intercept**: Set \( x = 0 \):
\[
g(0) = 0 + 1 = 1
\]
So, the y-intercept is \( (0, 1) \).
- **X-intercept**: Set \( g(x) = 0 \):
\[
x + 1 = 0 \implies x = -1
\]
So, the x-intercept is \( (-1, 0) \).
### Summary of intercepts:
- \( f(x) \) has:
- Y-intercept: \( (0, 1) \)
- X-intercept: \( (-1, 0) \)
- \( g(x) \) has:
- Y-intercept: \( (0, 1) \)
- X-intercept: \( (-1, 0) \)
### Analyzing the statements:
a. **The x- and y-intercepts are different for each function.**
- This is **false** because both functions have the same intercepts.
b. **The graphs intersect at (1, 2).**
- We can check by substituting \( x = 1 \) into both functions:
\[
f(1) = 1^3 + 1 = 2 \quad \text{and} \quad g(1) = 1 + 1 = 2
\]
- The graphs do intersect at \( (1, 2) \), so this is **true**.
c. **Both functions have an x-intercept of (-1, 0) and a y-intercept of (0, 1).**
- This is **true** since both functions have these intercepts.
d. **Both functions have a y-intercept of (-1, 0) and an x-intercept of (0, 1).**
- This is **false** because the y-intercept of \( f(x) \) and \( g(x) \) is \( (0, 1) \), not \( (-1, 0) \).
### Conclusion:
- Statements b and c are true, but considering the context, since the question asks which statements are true without specifying, the most explicit intersection result would be:
**b. The graphs intersect at (1, 2).**