Question

compares 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. both functions have an x-intercept of (-1, 0) and a y-intercept of (0, 1)
b. both functions have an y-intercept of (-1, 0) and a x-intercept of (0, 1)
c. the x- and y-intercepts are different for each function.
d. the graphs intersect at (1, 2)

Answer 1

To compare the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \) in terms of their intercepts, let's calculate the x-intercepts and y-intercepts for both functions.

### Finding the intercepts

1. **X-Intercepts**:
- For \( f(x) = 0 \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1.
\]
So, the x-intercept of \( f(x) \) is \( (-1, 0) \).

- For \( g(x) = 0 \):
\[
x + 1 = 0 \implies x = -1.
\]
So, the x-intercept of \( g(x) \) is also \( (-1, 0) \).

2. **Y-Intercepts**:
- For \( f(x) \), substitute \( x = 0 \):
\[
f(0) = 0^3 + 1 = 1.
\]
So, the y-intercept of \( f(x) \) is \( (0, 1) \).

- For \( g(x) \), substitute \( x = 0 \):
\[
g(0) = 0 + 1 = 1.
\]
So, the y-intercept of \( g(x) \) is also \( (0, 1) \).

### Conclusion about the intercepts
- Both functions \( f(x) \) and \( g(x) \) share the same x-intercept of \( (-1, 0) \) and the same y-intercept of \( (0, 1) \).

### Checking the statements:
- **a. both functions have an x-intercept of (-1, 0) and a y-intercept of (0, 1)** - **True**.
- **b. both functions have a y-intercept of (-1, 0) and a x-intercept of (0, 1)** - **False** (the x- and y-intercepts are incorrectly stated).
- **c. the x- and y-intercepts are different for each function.** - **False** (they are the same).
- **d. the graphs intersect at (1, 2)** - **False** (we have not determined this intersection yet).

### Finding intersection point (optional)
To find if they intersect at (1, 2), we check:
\[
f(1) = 1^3 + 1 = 2 \quad \text{and} \quad g(1) = 1 + 1 = 2.
\]
So the point (1, 2) is indeed on both graphs. However, this does not change the fact about the intercepts.

### **Final Answer:**
The correct statement is **a. both functions have an x-intercept of (-1, 0) and a y-intercept of (0, 1)**.