Question Type 1: Finding composite functions given two functions Exercises

Evaluate $(f\circ g)(1)$ for $f(x)=6x^2+3$ and $g(x)=x e^x$.

Let $f(x)=\sqrt{x}$ and $g(x)=x-3$. Find $(f\circ g)(x)$ and determine its domain.

Let $f(x)=e^x$ and $g(x)=3x^2$. Find both $(f\circ g)(x)$ and $(g\circ f)(x)$.

With the same functions $f(x)=6x^2+3$ and $g(x)=x e^x$, find $(g\circ f)(x)$.

Given $f(x)=6x^2+3$ and $g(x)=x e^x$, find $(f\circ g)(x)$.

Define $f(x)=\tfrac{1}{x+2}$ and $g(x)=x^2$. Compute and simplify $(g\circ f)(x)$, stating its domain.

Let $f(x)=\ln x$ and $g(x)=6x+5$. Express $(f\circ g)(x)$ and state its domain.

Given $h(x)=\ln(x^2+1)$ and $f(x)=x^3$, find $(h\circ f)(x)$ and $(f\circ h)(x)$.

Given $f(x)=\sqrt{x}$ and $g(x)=x^2-4x+3$, find $(f\circ g)(x)$ and determine its domain.

Let $f(x)=ax+b$ and $g(x)=cx+d$, where $a,b,c,d$ are constants. Find expressions for $(f\circ g)(x)$ and $(g\circ f)(x)$ in terms of $a,b,c,d$.

Let $f(x)=\frac{x}{1-x}$, $g(x)=x+4$, and $h(x)=2x-1$. Express $(f\circ g\circ h)(x)$ in simplest form.

Prove that $(f\circ g)^{-1}=g^{-1}\circ f^{-1}$ for $f(x)=2x-3$ and $g(x)=5x+1$ by explicit calculation.