Certainly! To solve this problem, we'll combine the functions by substituting [tex]\( x(y) = z \)[/tex] into the function [tex]\( f(z) = 6z + 5 \)[/tex] and simplify the expression accordingly.
1. Define the independent variable transformation:
Let's define [tex]\( x(y) \)[/tex]. Given [tex]\( x(y) = z \)[/tex], we assume:
[tex]\[
z = y
\][/tex]
2. Substitute [tex]\( x(y) \)[/tex] into [tex]\( f(z) \)[/tex]:
We are given [tex]\( f(z) = 6z + 5 \)[/tex]. We need to substitute [tex]\( z = y \)[/tex] into this function.
3. Rewrite [tex]\( f(z) \)[/tex] in terms of [tex]\( y \)[/tex]:
Since [tex]\( z = y \)[/tex], we can write:
[tex]\[
f(z) = f(y)
\][/tex]
4. Simplify the function:
Now substitute [tex]\( z \)[/tex] with [tex]\( y \)[/tex] in the function [tex]\( f(z) \)[/tex]:
[tex]\[
f(y) = 6y + 5
\][/tex]
5. Verify with an example value:
Let's choose a specific value for [tex]\( y \)[/tex] to check our result. Suppose [tex]\( y = 3 \)[/tex]:
[tex]\[
z = y = 3
\][/tex]
[tex]\[
f(z) = f(3)
\][/tex]
[tex]\[
f(3) = 6 \cdot 3 + 5 = 18 + 5 = 23
\][/tex]
Therefore, when [tex]\( y = 3 \)[/tex], we have [tex]\( z = 3 \)[/tex] and [tex]\( f(z) = 23 \)[/tex].
So, the simplified function after the substitution is:
[tex]\[
f(y) = 6y + 5
\][/tex]
With the example value [tex]\( y = 3 \)[/tex]:
[tex]\[
(3, 3, 23)
\][/tex]
This confirms that substituting [tex]\( x(y) = z \)[/tex] into [tex]\( f(z) = 6z + 5 \)[/tex] yields the simplified expression [tex]\( f(y) = 6y + 5 \)[/tex].
