Function

[aalpha]

Can anyone help me to solve the following problem. Of course its not home work problem.. thank you in advance..

f(x) = x² + 3

gf = 2x² + 3 find function g.

kindly explain me such type of problem..

 

[soroban]

Hello, aalpha!

\(\displaystyle f(x) \:=\: x^2+ 3\)
\(\displaystyle g(f(x))\:=\: 2x^2+ 3\)

\(\displaystyle \text{Find function }g(x).\)

I had to baby-talk my way through it . . .


We want a function \(\displaystyle g(x)\) which will change \(\displaystyle x^2+3\) to \(\displaystyle 2x^2+3\)


We have: .\(\displaystyle f(x) \:= \: x^2+3\)

Subtract 3: .\(\displaystyle f(x) - 3 \:=\: x^2\)

Multiply by 2: .\(\displaystyle 2[f(x)-3] \:=\:2x^2\)

Add 3: .\(\displaystyle 2[f(x) - 3] + 3 \:=\:2x^2+3\)


Hence, the \(\displaystyle g\)-function is: "subtract 3, multiply by 2, add 3".


Therefore: .\(\displaystyle g(x) \;=\;2(x-3) + 3 \;=\;2x-3\)