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) Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).
B) The x- and y-intercepts are different for each function.
C) The graphs intersect at (1,2).
D) Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).
A) Both functions have a y-intercept of (−1,0) and an x-intercept of (0,1).
B) The x- and y-intercepts are different for each function.
C) The graphs intersect at (1,2).
D) Both functions have an x-intercept of (−1,0) and a y-intercept of (0,1).
Answers
GPT 3.5
The y-intercept is the point where the graph of a function intersects the y-axis. To find the y-intercept, we set x = 0 and evaluate the function.
For f(x) = x^3 + 1, when x = 0, we have f(0) = 0^3 + 1 = 1. Therefore, the y-intercept of f(x) is (0, 1).
For g(x) = x + 1, when x = 0, we have g(0) = 0 + 1 = 1. Therefore, the y-intercept of g(x) is (0, 1).
Thus, the y-intercepts of both functions are the same, (0, 1).
The x-intercept is the point where the graph of a function intersects the x-axis. To find the x-intercept, we set y = 0 and solve for x.
For f(x) = x^3 + 1, when y = 0, we have x^3 + 1 = 0. There is no real solution to this equation, so f(x) does not have an x-intercept.
For g(x) = x + 1, when y = 0, we have x + 1 = 0. Solving for x, we find x = -1. Therefore, the x-intercept of g(x) is (-1, 0).
Thus, the x-intercept for g(x) is (-1, 0), while f(x) does not have an x-intercept.
Therefore, the correct statement about the intercepts of the functions is:
B) The x- and y-intercepts are different for each function.
For f(x) = x^3 + 1, when x = 0, we have f(0) = 0^3 + 1 = 1. Therefore, the y-intercept of f(x) is (0, 1).
For g(x) = x + 1, when x = 0, we have g(0) = 0 + 1 = 1. Therefore, the y-intercept of g(x) is (0, 1).
Thus, the y-intercepts of both functions are the same, (0, 1).
The x-intercept is the point where the graph of a function intersects the x-axis. To find the x-intercept, we set y = 0 and solve for x.
For f(x) = x^3 + 1, when y = 0, we have x^3 + 1 = 0. There is no real solution to this equation, so f(x) does not have an x-intercept.
For g(x) = x + 1, when y = 0, we have x + 1 = 0. Solving for x, we find x = -1. Therefore, the x-intercept of g(x) is (-1, 0).
Thus, the x-intercept for g(x) is (-1, 0), while f(x) does not have an x-intercept.
Therefore, the correct statement about the intercepts of the functions is:
B) The x- and y-intercepts are different for each function.