To determine the [tex]\( x \)[/tex]-intercept of the original function [tex]\( f(x) = 2x - \frac{1}{2} \)[/tex], we follow these steps:
1. Understand the [tex]\( x \)[/tex]-intercept: The [tex]\( x \)[/tex]-intercept is the value of [tex]\( x \)[/tex] where the function [tex]\( f(x) \)[/tex] equals zero. This means we need to solve the equation [tex]\( f(x) = 0 \)[/tex].
2. Set the function equal to zero:
[tex]\[
f(x) = 2x - \frac{1}{2} = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
- First, isolate [tex]\( x \)[/tex] by adding [tex]\( \frac{1}{2} \)[/tex] to both sides of the equation:
[tex]\[
2x - \frac{1}{2} + \frac{1}{2} = 0 + \frac{1}{2}
\][/tex]
which simplifies to:
[tex]\[
2x = \frac{1}{2}
\][/tex]
- Next, divide both sides by 2:
[tex]\[
x = \frac{\frac{1}{2}}{2} = \frac{1}{4}
\][/tex]
Thus, the [tex]\( x \)[/tex]-intercept for [tex]\( f(x) = 2x - \frac{1}{2} \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
Comparison with the given statements:
- The [tex]\( x \)[/tex]-intercept is NOT the constant in the [tex]\( f(x) \)[/tex] equation, since the constant in [tex]\( f(x) \)[/tex] is [tex]\(-\frac{1}{2}\)[/tex].
- The [tex]\( x \)[/tex]-intercept is NOT the constant in the [tex]\( f^{-1}(x) \)[/tex] equation, as the constant in [tex]\( f^{-1}(x) \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
- The correct statement is:
[tex]\[
\text{The } x\text{-intercept for } f(x) \text{ is the reciprocal of the constant in the } f(x) \text{ equation.}
\][/tex]
since [tex]\((2)^{-1} = \frac{1}{2}\)[/tex]. However, we need to solve for [tex]\( x \)[/tex] by dividing by 2, leading to [tex]\( \frac{1}{4} \)[/tex].
So the correct statement is: The x-intercept for [tex]\( f(x) \)[/tex] is the reciprocal of the constant in the [tex]\( f(x) \)[/tex] equation divided by 2.
