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=(1-n)/(5) \end{gathered}`

The answer is:

`m=(1-n)/(5)`