Question
What is the range of the function f(x) = x²+60x?
All real numbers that are greater than or equal to 0 and less than or equal to 30
All real numbers that are greater than or equal to 0 and less than or equal to 60
All real numbers that are greater than or equal to 900
All real numbers that are less than or equal to 900
All real numbers that are greater than or equal to 0 and less than or equal to 30
All real numbers that are greater than or equal to 0 and less than or equal to 60
All real numbers that are greater than or equal to 900
All real numbers that are less than or equal to 900
Answers
GPT-4o mini
To determine the range of the function \( f(x) = x^2 + 60x \), we can rewrite it in a form that makes it easier to analyze.
We can complete the square:
\[
f(x) = x^2 + 60x = (x + 30)^2 - 900
\]
This means that the function \( f(x) \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive). The vertex of this parabola, which gives us the minimum value, occurs at \( x = -30 \):
\[
f(-30) = (-30 + 30)^2 - 900 = 0 - 900 = -900
\]
Since the parabola opens upwards from this vertex, the range of \( f(x) \) is:
\[
f(x) \geq -900
\]
This means that the range of the function is:
\[
[-900, \infty)
\]
None of the multiple-choice options provided match this range. However, the focus should be on the minimum value of \( f(x) \), which is \(-900\), indicating that all outputs of the function are at least \(-900\) and can go up to infinity.
We can complete the square:
\[
f(x) = x^2 + 60x = (x + 30)^2 - 900
\]
This means that the function \( f(x) \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive). The vertex of this parabola, which gives us the minimum value, occurs at \( x = -30 \):
\[
f(-30) = (-30 + 30)^2 - 900 = 0 - 900 = -900
\]
Since the parabola opens upwards from this vertex, the range of \( f(x) \) is:
\[
f(x) \geq -900
\]
This means that the range of the function is:
\[
[-900, \infty)
\]
None of the multiple-choice options provided match this range. However, the focus should be on the minimum value of \( f(x) \), which is \(-900\), indicating that all outputs of the function are at least \(-900\) and can go up to infinity.