IB Mathematics Analysis and Approaches Question Type 2: Finding Composite Functions of Two More Complex Functions Exercises

Let $f(x)=\ln x$ and $g(x)=x^x$ for $x>0$. Find $(f\circ g)(x)$ and simplify.

Let $f(x)=e^{2x}$ and $g(x)=\ln (x^2+1)$. Find $(f\circ g)(x)$ in simplest form.

Let $f(x)=\frac{1}{x-2}$ and $g(x)=\frac{x+1}{x-1}$. Find $(f\circ g)(x)$.

Let $f(x)=\sqrt{3x+1}$ and $g(x)=x^2-5$. Find $(f\circ g)(x)$ and simplify.

Let $f(x)=\ln (1+x)$ and $g(x)=\sqrt{2x-1}$. Find $(g\circ f)(x)$ and simplify.

Let $f(x)=2x-3$ and $g(x)=x^2+4x$. Find $(g\circ f)(x)$.

Let $f(x)=\frac{1}{\sqrt{x}}$ and $g(x)=\frac{2x+3}{x}$ for $x\ne 0$. Find $(f\circ g)(x)$ in simplest radical form.

Let $f(x)=\frac{x+2}{x-2}$ and $g(x)=\sqrt{x^2-1}$. Find $(f\circ g)(x)$.

Let $f(x)=\cos (\ln x)$ for $x>0$ and $g(x)=e^{x^2}$. Find $(f\circ g)(x)$.

Let $f(x)=5x^3+9$, $g(x)=x{5}^x{e}^x$, and $h(x)=3g(x)f(g(x))$. Find $(h\circ f)(x)$ in closed form.

Let $f(x)=\frac{3}{1-x}$ and $g(x)=\frac{x}{x-3}$. Find $(g\circ f)(x)$ and simplify fully.

Let $f(x)=\sin (3x)$ and $g(x)=x^2-\pi$. Find $(f\circ g)(x)$ and simplify using trigonometric identities.