<< Chapter < Page
  Functions   Page 1 / 1
Chapter >> Page >

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.

Function definition and inequality

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 x = log a 3 x 2 x + 4

Here, logrithmic function is defined for a 0,1 1, and

3 x 2 x + 4 > 0

2 : f x = arcsin 3 x 2 x + 4

Here, arcsine function is defined in the domain [-1,1]. Hence,

- 1 3 x 2 x + 4 1

It is clear that we need to have clear understanding of algebraic inequalities as function definitions are defined with certain condition(s).

Forms of function inequality

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 - 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

Inequalities

Some important definitions/ results are enumerated here :

  • Inequalities involve a relation between two real numbers or algebraic expressions.
  • The inequality relations are "<", ">", "≤" and "≥".
  • Equal numbers can be added or subtracted to both sides of an inequality.
  • Both sides of an inequality can be multiplied or divided by a positive number without any change in the inequality relation.
  • Both sides of an inequality can be multiplied or divided by a negative number with reversal of inequality relation.
  • Both sides of an inequality can be squared, provided expressions are non-negative. As a matter of fact, this conclusion results from rule that we can multiply both sides with a positive number.
  • When both sides are replaced by their inverse, the inequality is reversed .

Equivalently, we may state above deductions symbolically.

If x > y , then :

x + a > y + a

a x > a y ; a > 0

a x < a y ; a < 0

x 2 > y 2 ; x , y > 0

1 x < 1 y ; when “x” and “y” have same sign.

It is evident that we can deduce similar conclusions with the remaining three inequality signs.

Intervals with inequalities

In general, a continuous interval is denoted with "less than (<)" or "less than equal to (≤)" inequalities like :

1 < x 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 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 - 3

Two disjointed intervals are combined with “union” operator like :

1 < x 2 x > 5

Linear inequality

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 x = log e 3 x + 4

The argument of logarithmic function is a positive number. Hence,

x > - 4 3

Therefore, interval of x i.e. domain of logarithmic function is - 4 / 3, . The figure shows the values of “x” on a real number line as superimposed on x-axis. Note x= - 4/3 is excluded.

Graph of logarithmic function

Domain is traced on x-axis.

When f(x) = 0,

3 x + 4 = e f x = e 0 = 1

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.

We shall similarly consider inequalities involving polynomials of higher degree, rational function etc in separate modules.

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 x = 2 x + 2 = 0

x = - 1

At x=0,

f x = 2

We draw a line passing through these two points as shown in the figure. From the figure, we conclude that :

Graph of linear function

Graph is continuous for all values of x.

f x < 0 ; x - , - 1

f x 0 ; x ( - , - 1 ]

f x > 0 ; x - 1,

f x 0 ; x [ - 1, )

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask