To find the reflection of the function [tex]\( f(x) = \frac{1}{2} x - 3 \)[/tex] in the [tex]\( x \)[/tex]-axis, we need to perform the following steps:
1. Understand the Concept of Reflection in the [tex]$x$[/tex]-Axis:
- Reflecting a function across the [tex]\( x \)[/tex]-axis means that we take the negative of the function's output value. In other words, if the original function is [tex]\( f(x) \)[/tex], then the reflected function [tex]\( g(x) \)[/tex] will be [tex]\( g(x) = -f(x) \)[/tex].
2. Apply the Reflection to the Given Function:
- Start with the original function: [tex]\( f(x) = \frac{1}{2} x - 3 \)[/tex].
- Reflect this function by multiplying it by [tex]\(-1\)[/tex]:
[tex]\[
g(x) = - \left( \frac{1}{2} x - 3 \right)
\][/tex]
3. Distribute the Negative Sign:
- Distribute the [tex]\(-1\)[/tex] across each term in the function:
[tex]\[
g(x) = - \frac{1}{2} x + 3
\][/tex]
- which can be rewritten as:
[tex]\[
g(x) = 3 - \frac{1}{2} x
\][/tex]
4. State the Reflected Function:
- Therefore, the reflected function is:
[tex]\[
g(x) = 3 - \frac{1}{2} x
\][/tex]
5. Verify the Result by Comparison to the Formula:
- Finally, we compare the original and reflected functions:
[tex]\[
f(x) = \frac{1}{2} x - 3
\][/tex]
[tex]\[
g(x) = 3 - \frac{1}{2} x
\][/tex]
In conclusion, the reflection of the function [tex]\( f(x) = \frac{1}{2} x - 3 \)[/tex] across the [tex]\( x \)[/tex]-axis is [tex]\( g(x) = 3 - \frac{1}{2} x \)[/tex].
