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Families of Functions

6.4 Systems with a Single Parameter

System of Equations with a Single Parameter

  • We have seen how simultaneous equations look like. For instance,

\left\{\begin{array}{ll}6=x+y+z & (1) \\ 4=4 x+2 y+z & (2)\end{array}\right.

and how there can be unique, infinite or no solutions for each set of simultaneous equations. These are explored in section “Solving Simultaneous Linear Equations” in A2 – Algebra.

  • Literal equations are very similar, just that instead of just numbers, we will have unknown constants. Such as,

\left\{\begin{array}{ll}x+y=2 m+3 & (1) \\ 3 x-2 y=7 m-1 & (2)\end{array}\right.

where m is a constant. Hence the solution (x,\ y) will be expressed in terms of m, and again there are unique, infinite, or no solutions. This will depend on the value(s) of m.

  • Parameters are a little different from literals. They are technically speaking, variables and not constants. It can be seen as a link between two (or more) variables.
  • For instance, take a cuboid with side length x for example. We denote the surface area as S, volume as V, then we have
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6.3 Families of Functions and Solving Literal Equations

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).

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