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).