Practice section 1 of 2

Complete the following table.

The function $f$ is defined by $f:x\mapsto3x^2+1$, $x\in\mathbb R$. Find (a) $f(3)$, (b) $f(5x)$, (c) $f(2y+1)$.

The function $f$ is defined by $f(x)=x^2-5x$, $x\in\mathbb R$. Find (a) $f(x+2)$, (b) $f(x^2)$, (c) $[f(x)]^2$.

State and justify which of the following is a function. (a) $f:x\mapsto x^2+1$, $x\in\mathbb R$. (b) $g:x\mapsto\pm\sqrt{x}$, $x\in\mathbb R$, $x\ge0$.

Determine the range of each of the following functions. (a) $f:x\mapsto\ln x-x$, $x\in\mathbb R$, $1<x\le e^2$. (b) $g:x\mapsto(x+2)^2+e^x$, $x\in\mathbb R$, $x>-4$.

Given that $f(x)=\begin{cases}x^2&0\le x<2,\\7-x&2\le x<3.\end{cases}$ Sketch the graph of $y=f(x)$ and state the range of the function.