Question

f(x) = x³ +1 g(x) = x + 1 X -2-7 -10 -1 0 1 1 2 2 1 0 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) Both functions have an x-intercept of (-1,0) and a y-intercept of (0, 1). The graphs intersect at (1, 2) Both functions have a y-intercept of (-1,0) and an x-intercept of (0, 1). The x- and y-intercepts are different for each function

Answer 1

To compare the functions \( f(x) = x^3 + 1 \) and \( g(x) = x + 1 \) and examine their intercepts, we need to find the x-intercepts and y-intercepts of both functions.

### Finding Intercepts

1. **Y-Intercepts**:
- The y-intercept occurs when \( x = 0 \):
- For \( f(x) \):
\[
f(0) = 0^3 + 1 = 1 \quad \text{(y-intercept is (0, 1))}
\]
- For \( g(x) \):
\[
g(0) = 0 + 1 = 1 \quad \text{(y-intercept is (0, 1))}
\]

2. **X-Intercepts**:
- The x-intercept occurs when \( y = 0 \) (i.e., set the functions to 0 and solve for \( x \)):
- For \( f(x) \):
\[
x^3 + 1 = 0 \implies x^3 = -1 \implies x = -1 \quad \text{(x-intercept is (-1, 0))}
\]
- For \( g(x) \):
\[
x + 1 = 0 \implies x = -1 \quad \text{(x-intercept is (-1, 0))}
\]

### Summary of Intercepts
- Both functions have:
- An x-intercept of (-1, 0)
- A y-intercept of (0, 1)

### Comparing with Statements
Now let's compare what we found with the statements provided:

1. **Both functions have an x-intercept of (-1,0) and a y-intercept of (0, 1).** (True)
2. **The graphs intersect at (1, 2).** (We need to check this point, but first, we can see it's not a statement regarding intercepts.)
3. **Both functions have a y-intercept of (-1,0) and an x-intercept of (0, 1).** (False, as shown above.)
4. **The x- and y-intercepts are different for each function.** (False, as shown above.)

### Conclusion
The statement that is true about the intercepts of the functions is:

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