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Inequality is an important concept in understanding function and its properties – particularly domain and range. Many function forms are valid in certain interval(s) of real numbers. This means definition of function is subjected to certain restriction of values with respect to dependent and independent variables. The restriction is generally evaluated in terms of algebraic inequalities, which may involve linear, quadratic, higher degree polynomials or rational polynomials.
A function imposes certain limitations by virtue of definition itself. We have seen such restriction with respect to radical functions in which polynomial inside square root needs to be non-negative. We have also seen that denominator of a rational function should not be zero. We shall learn about different functions in subsequent modules. Here, we consider few examples for illustration :
1 : $f\left(x\right)=\mathrm{log}{}_{a}\left(3{x}^{2}-x+4\right)$
Here, logrithmic function is defined for $a\in \left(\mathrm{0,1}\right)\cup \left(\mathrm{1,}\infty \right)$ and
$$3{x}^{2}-x+4>0$$
2 : $f\left(x\right)=\mathrm{arcsin}\left(3{x}^{2}-x+4\right)$
Here, arcsine function is defined in the domain [-1,1]. Hence,
$$-1\le 3{x}^{2}-x+4\le 1$$
It is clear that we need to have clear understanding of algebraic inequalities as function definitions are defined with certain condition(s).
Function inequality compares function to zero. There are four forms :
1 : f(x)<0
2 : f(x) ≤ 0
3 : f(x)>0
4 : f(x) ≥ 0
Here, f(x)<0 and f(x)>0 are strict inequalities as they confirm the notion of “less than” and “greater than”. There is no possibility of equality. If a strict inequality is true, then non-strict equality is also true i.e.
1 : If f(x)>0 then f(x) ≥ 0 is true.
2 : If f(x) ≥ 0 then f(x)>0 is not true.
3 : If f(x)<0 then f(x) ≤ 0 is true.
4 : If f(x) ≤ 0 then f(x)<0 is not true.
Further, we may be presented with inequality which compares function to non-zero value :
$$3{x}^{2}-x\le -4$$
However, such alterations are equivalent expressions. We can always change this to standard form which compares function with zero :
$$3{x}^{2}-x+4>0$$
Some important definitions/ results are enumerated here :
Equivalently, we may state above deductions symbolically.
$$\text{If}\phantom{\rule{1em}{0ex}}x>y,\phantom{\rule{1em}{0ex}}\text{then :}\phantom{\rule{1em}{0ex}}$$
$$x+a>y+a$$
$$ax>ay;\phantom{\rule{1em}{0ex}}a>0$$
$$ax<ay;\phantom{\rule{1em}{0ex}}a<0$$
$${x}^{2}>{y}^{2};x,y>0$$
$$\frac{1}{x}<\frac{1}{y};\phantom{\rule{1em}{0ex}}\text{when \u201cx\u201d and \u201cy\u201d have same sign.}$$
It is evident that we can deduce similar conclusions with the remaining three inequality signs.
In general, a continuous interval is denoted with "less than (<)" or "less than equal to (≤)" inequalities like :
$$1<x\le 5$$
The segment of a real number line from a particular number extending to plus infinity is denoted with “greater than” or “greater than equal to” inequalities like :
$$x\ge 3$$
The segment of real number line from minus infinity to a certain number on real number line is denoted with “less than(<) or less than equal to (≤)” inequalities like :
$$x\le -3$$
Two disjointed intervals are combined with “union” operator like :
$$1<x\le 2\phantom{\rule{1em}{0ex}}\cup \phantom{\rule{1em}{0ex}}x>5$$
Linear function is a polynomial of degree 1. A linear inequality can be solved for intervals of valid “x” and “y” values, applying properties of inequality of addition, subtraction, multiplication and division. For illustration, we consider a logarithmic function, whose argument is a linear function in x.
$$f\left(x\right)={\mathrm{log}}_{e}\left(3x+4\right)$$
The argument of logarithmic function is a positive number. Hence,
$$\Rightarrow x>-\frac{4}{3}$$
Therefore, interval of x i.e. domain of logarithmic function is $\left(-4/\mathrm{3,}\infty \right)$ . The figure shows the values of “x” on a real number line as superimposed on x-axis. Note x= - 4/3 is excluded.
When f(x) = 0,
$$3x+4={e}^{f\left(x\right)}={e}^{0}=1$$
$$\Rightarrow x=-1$$
It means graph intersects x-axis at x=-1 as shown in the figure. From the figure, it is clear that range of function is real number set R.
Problem : A linear function is defined as f(x)=2x+2. Find valid intervals of “x” for each of four inequalities viz f(x)<0, f(x) ≤ 0, f(x)>0 and f(x) ≥ 0.
Solution : Here, given function is a linear function. At y=0,
$$f\left(x\right)=2x+2=0$$
$$\Rightarrow x=-1$$
At x=0,
$$f\left(x\right)=2$$
We draw a line passing through these two points as shown in the figure. From the figure, we conclude that :
$$\Rightarrow f\left(x\right)<0;\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,-1\right)$$
$$\Rightarrow f\left(x\right)\le 0;\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,-1\right]$$
$$\Rightarrow f\left(x\right)>0;\phantom{\rule{1em}{0ex}}x\in \left(-\mathrm{1,}\infty \right)$$
$$\Rightarrow f\left(x\right)\ge 0;\phantom{\rule{1em}{0ex}}x\in \left[-\mathrm{1,}\infty \right)$$
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