First Principles AI

(a) If $f(x)=x^2-1$ , find $f(-3)$ .

(a) If $f(x)=x^2-1$ , find $f(0)$ .

(a) If $f(x)=x^2-1$ , find $f(x-2)$ .

(iv)

(a) If $f(x)=x^2-1$ , find $f(x^2+3)$ .

(v)

(b) If $f(x)=\sqrt{2x+3}$ , find $f(-3)$ .

(vi)

(b) If $f(x)=\sqrt{2x+3}$ , find $f(0)$ .

(vii)

(b) If $f(x)=\sqrt{2x+3}$ , find $f(x-2)$ .

(viii)

(b) If $f(x)=\sqrt{2x+3}$ , find $f(x^2+3)$ .

Question 2

Find $\dfrac{f(a+h)-f(a)}{h}$ and simplify where:

$f(x)=4x+7$

$f(x)=\sin x$

$f(x)=x^2+x^2-1$

$f(x)=\tan x$

Question 3

Express the following:

(a)

The area of a square as a function of its perimeter $P$ .

(b)

The circumference $C$ of a circle as a function of its area $A$ .

(c)

The surface area $S$ of a cube as a function of its volume $V$ .

Question 4

Find the domain and the range of the function $g$ defined below:

$g(x)=5-x$

$g(x)=\sqrt{x+2}$

$g(x)=\begin{cases}6x+7,&x\le -2\\4-3x,&x>-2\end{cases}$

(iv)

$g(x)=|x-5|$

(v)

$g(x)=\dfrac{x+2}{3-x}$

Question 5

Given $f(x)=x^3-ax^2+bx+1$ , if $f(2)=-3$ and $f(-1)=0$ , find the values of $a$ and $b$ .

Question 6

A stone falls from a height of 60m on the ground, the height $h$ after $x$ seconds is approximately given by $h(x)=40-10x^2$ .

What is the height of stone when $x=1$ sec, $x=1.5$ sec, and $x=1.7$ sec?

When does the stone strike the ground?

Question 7

Consider the function $f(x)=3x-5$ .

Determine the domain and range of $f(x)$ .

Is the function one-to-one? Justify your answer.

Is the function onto if the co-domain is all real numbers? Explain.

Question 8

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=\dfrac{2x-3}{x+1}$ .

Find the domain and range of $f(x)$ .

Determine whether $f(x)$ is onto.

Prove that $f(x)$ is one-to-one.

Question 9

Consider the function $f:\mathbb{R}^+\to\mathbb{R}^+$ defined by $f(x)=e^x$ . Show that $f$ is bijective.

Question 10

Let $g:\mathbb{R}\to\mathbb{R}$ be given by $g(x)=x^2-3x$ . Determine if $g$ is injective and/or surjective.