Question Type 1: Finding composite functions given two functions Exercises

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

Express $(f \circ g)(x)$ and state its domain.

Find $(g \circ f)(x)$.

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

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.

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

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

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

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

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