Final answer:
To determine the function rule for y=r(x), we know that (r∘s)(x)=r(s(x))=6x−10 when s(x)=2x−4. We find the rule of r(x) by solving for a function that takes (2x−4) and outputs (6x−10), which results in the rule r(x)=3x−2.
Step-by-step explanation:
The question involves composition of functions and requires us to find the rule for the function y=r(x) knowing s(x)=2x−4 and (r∘s)(x)=6x−10.
The composition of functions (r∘s)(x) means that we first apply s(x) and then apply r(x) to the result.
Therefore, the resulting composition is r(s(x)). Given that s(x)=2x−4 and (r∘s)(x)=6x−10, we substitute s(x) into the composition to get r(2x−4) = 6x−10. To find the function rule for y=r(x), we have to express r(x) in terms of x alone without the composition.
This means we should solve for r(x) assuming that r(x) transforms input 2x−4 into 6x−10.
To solve for r(x), we set x to (2x−4) and then determine the value of r(x) that would yield 6x−10 when x is replaced by 2x−4.
Upon substitution, we find that r(2x−4) must increase every x value by a factor of 3 and then subtract an additional 2 to yield 6x−10.
Hence, if r(x) accounts for the input x alone, the rule must be:
r(x) = 3x − 2