1. Find f(g)(x) when f(x)=x^2 - 7x +12 and g(x)= 7/x^2 - 16.
idk Max, I think this is the answer...
\(f(g(x)) \;\;or\;\; (f\cdot g)(x)\quad when\\ f(x)=x^2-7x+12 \quad and \quad g(x)=\frac{7}{x^2}-16\\~\\ \begin{align} (f\cdot g)(x)&=(\frac{7}{x^2}-16)^2-7(\frac{7}{x^2}-16)+12\\ &=(\frac{49}{x^4}-[2*\frac{7}{x^2}*16]+256)-(\frac{49}{x^2}-112)+12\\ &=\frac{49}{x^4}-\frac{224}{x^2}+256-\frac{49}{x^2}+112+12\\ &=\frac{49}{x^4}-\frac{273}{x^2}+380\\ \end{align} \)
2. Find the inverse function for f(x)=(sqrt2x-6).
When you do part 2 max you must be very careful of the domain.
f(x)=(sqrt2x-6).
\(y=\sqrt{2x}-6\)
Straight away I see that x must be greater or equal to 0 and y must be greater than or equal to -6
so when you take the inverse x>=-6 and y>=0
\(f(x)=\sqrt{2x}-6\\ y=\sqrt{2x}-6\\ y+6=\sqrt{2x}\\ (y+6)^2=2x\\ x=\frac{(y+6)^2}{2}\\ \text{the inverse function becomes}\\ f^{-1}(x)=\frac{(x+6)^2}{2} \qquad where \quad x\ge-6\)
Here are the 2 graphs drawn with Desmos graphing calculator.
NOTE that the invers of a function is the reflection of it across the line y=x. It is important that you understand and remember this.
