Question

Given the function k(x)=mx+1, h(x)=3x-5 and kh(x)=3mx+n. Express m in terms of n.

Answer 1

The functions are given as shown:

$\begin{gathered} k(x)=mx+1 \\ h(x)=3x-5 \end{gathered}$

The question also provides:

$kh(x)=3mx+n$

The expression kh(x) represents a composite of functions such that:

$kh(x)=k(h(x))$

Let us evaluate the value of the composite of the functions:

$\begin{gathered} k(h(x))=m(3x-5)+1 \\ k(h(x))=3mx-5m+1 \end{gathered}$

Therefore, we can equate the given value of kh(x) and the derived one:

$3mx+n=3mx-5m+1$

Hence, we can solve for m in the equation above by collecting like terms on opposite ends of the equality sign and get our answer:

$\begin{gathered} 3mx-3mx+5m=1-n \\ 5m=1-n \\ m=\frac{1-n}{5} \end{gathered}$

The answer is:

$m=\frac{1-n}{5}$