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Precalculus Problems

Precalculus prepares students for calculus. It is a group of topics that are required to study before getting started with calculus. Precalculus is usually taught from grade 7 to 9. It covers very important topics which are not only useful in the field of calculus, but also in various situations of real life.

The main topics covered during the study of precalculus are:

  1.    Number system
  2.    Set theory
  3.    Functions
  4.    Polynomials
  5.    Trigonometry
  6.    Conic sections
  7.    Limits and continuity
  8.    Vector algebra
  9.    Logarithmic functions
10.    Binomial theorem
11.    Graphical representation of functions
12.    Matrices and determinants
Few problems based on precalculus course are illustrated below:

Function Composition Problem

Problem 1:


Consider two functions $f(x)=x^{2}$ and $g(x)=x+1$. Determine whether f o g = g o f?

Solution:


f o g = f {g(x)}
= f (x + 1)
= $(x+1)^{2}$
= $x^{2}+1+2x$

g o f = g {f(x)}
= g($x^{2}$)
= $x^{2}+1$

Therefore, we conclude that $fog\neq gof$

Set Theory Problem

Problem 2:


Consider two sets A = {x : x is an even number, $2\leq x\leq 8$, $x\in N$} and B = {x : $x< 7$, $x\in N$}. Find $A\cup B$, $A\cap B$ and $B\cap A$. Determine whether $A\cap B=B\cap A$.

Solution:


By the given definition of sets, we get
A = {2, 4, 6, 8} and B = {1, 2, 3, 4, 5, 6}
$A\cup B$ = {1, 2, 3, 4, 5, 6, 8}
$A\cap B$ = {2, 4, 6}
$B\cap A$ = {2, 4, 6}

We conclude that,
$A\cap B = B\cap A$

Trigonometry Problem

Problem 3:

Two sides of a right $\bigtriangleup ABC$ are given in the following figure:
Precalculus Problems
Calculate the measure of $\angle C$.

Solution:
 

We know that:

$\sin C$ = $\frac{\text{Perpendicular}}{\text{Hypotenuse}}$

In $\bigtriangleup ABC$:

$\sin C$ = $\frac{AB}{AC}$

$\sin C$ = $\frac{\sqrt{3}}{2}$

$\sin C =\sin 60^{\circ}$
$\angle C= 60^{\circ}$

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