Solving Literal Equations

6.3 Families of Functions and Solving Literal Equations

For instance, $y=2 x^{3},\ y=-\frac{1}{4} x^{3}$ can all belong to a family function $f(x)=k x^{3},\ k \neq 0$ .

Functions from the same family would exhibit some similarities. Apart from the general shape of its graph, they might share some other things, such as a same intercept, a same asymptote, and so on.

Literal equations can also be applied on the 3 types of functions we learnt for various things; it is simply a combination of all the knowledges you have learnt.

Example

Let $f: R \rightarrow R$ , and $f(x)=m x+2$ where $m \in R^{+}$ . Find:

a) Its x-intercept;

b) The values of $m$ where its x-intercept is lesser than 2;

c) Its inverse function;

d) The equation of the line perpendicular to $f(x)$ at $(0,\ 2)$ .

Simply put, you just need to treat all pronumerals as numbers. Then, use your knowledge with polynomials to solve them.

Example

Solve the following:

a) $x^{2}+k x+k=0$ ;

Using our knowledge of solution for quadratic functions, we know that

In previous years, you are already equipped with the skills to solve simple linear equations like $4 y-\frac{(3 y+4)}{2}+\frac{1}{3}=\frac{5(4-y)}{3}$ , including their corresponding worded problems. The skills learnt are crucial and will be applied in the topics explored in this course.

A literal equation in $x$ is an equation whose solution will be expressed in terms of pronumerals rather than numbers.

In simpler terms, instead of solving

$2x+5=7$

instead you will be dealing with

$ax+b=c$