To solve this problem, we need to determine the correct composition function that represents using the coupon and then receiving the markdown.
1. Understanding the Functions:
- [tex]\( c(x) = x - 25 \)[/tex]: This represents the cost of the shoes after applying the \$25 coupon.
- [tex]\( s(x) = 0.75x \)[/tex]: This represents the cost of the shoes after a [tex]\(25\%\)[/tex] markdown.
2. Applying the Coupon First, Then the Markdown:
To find the composition function representing using the coupon first and then applying the markdown, we need to apply the markdown function [tex]\( s(x) \)[/tex] to the cost after the coupon function [tex]\( c(x) \)[/tex].
3. Step-by-Step Composition:
- Start by applying the coupon [tex]\( c(x) = x - 25 \)[/tex] to the original price [tex]\( x \)[/tex].
[tex]\[
\text{New price after coupon is } c(x) = x - 25.
\][/tex]
- Next, apply the markdown [tex]\( s(x) = 0.75x \)[/tex] to the result [tex]\( (x - 25) \)[/tex].
[tex]\[
s(c(x)) = 0.75(x - 25).
\][/tex]
4. Simplify the Composition:
- Distribute the [tex]\( 0.75 \)[/tex] inside the parentheses:
[tex]\[
s(c(x)) = 0.75 \cdot x - 0.75 \cdot 25.
\][/tex]
- Calculate [tex]\( 0.75 \times 25 \)[/tex]:
[tex]\[
0.75 \cdot 25 = 18.75.
\][/tex]
- Therefore, the expression simplifies to:
[tex]\[
s(c(x)) = 0.75x - 18.75.
\][/tex]
Thus, the composition function that represents using the coupon first and then receiving the markdown is:
[tex]\[
s(c(x)) = 0.75x - 18.75.
\][/tex]
Final Answer:
[tex]\[
\boxed{s(c(x)) = 0.75x - 18.75}
\][/tex]
