Join the IDNLearn.com community and start finding the answers you need today. Join our Q&A platform to receive prompt and accurate responses from knowledgeable professionals in various fields.
Reflect the function [tex]$f(x)=\frac{1}{2}x-3$[/tex] in the [tex]x[/tex]-axis.
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].
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.
