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Classification of Functions

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Constant Function

 A function which does not change as its parameters vary i.e., the function whose rate of change is zero. In short, a constant function is a function that always gives or returns the same value.

Or, let k be a constant, then function f (x) = k, ∀ x $\in$ R is known as constant function. 

Domain of f(x) = R and Range of f(x) = {k}. 

Polynomial Function 

The function y = f(x) = $a_0x^n+a_1x^{n-1}+...+n$ where $a_1$, $a_2$, $a_3$, ..., $a_n$ are real coefficients and n is a non-negative integer, is known as a polynomial function. If $a_0 \neq 0$, then degree of polynomial function is n. 

Domain of f(x) = R.

On range varies from function to function.

Rational Function

 If P (x) and Q(x) are polynomial functions, Q(x) $\neq$ 0, then function f(x) = $\frac{P(x)}{Q(x)}$ is known as rational function. 

Domain of f(x) = R - {x : Q(x) = 0} and range varies from function to function.

Irrational Function

The function containing one or more terms having non-integral rational powers of x are called irrational function. e.g., y = f(x) = $\frac{5x^{3/2}-7x^{1/2}}{x^{1/2}-1}$ 

Domain = varies from function to function.

Identity Function

Function f(x)= x, ∀ x $\in$ R is known as identity function. It is straight line passing through origin and having slope unity. 

Domain of f (x) = R and Range of f(x) = R

Square Root Function

The function that associates every positive real number x to +$\sqrt{x}$ is called the square root function, i.e., f(x) = +$\sqrt{x}$. 

Range of f(x) = [0, ∞).

Exponential Function

A function of the form f (x) = $a^x$ is a positive real number, is an exponential function. The value of the function depends upon the value of a for 0 < a < 1 , function is decreasing and for a > 1, function is increasing. 

Domain of f(x) = R and Range of f(x) = [0, ∞).

For Fig.a
Since a > 1, f is an increasing function. The graph of f is a curve which goes upward when x increases [i.e., f (x) increases when x increases] and goes downwards when x decreases [i.e., f (x) decreases when x decreases]. 

Also, since a > 1, a is positive and hence $a^x$ is positive for all x. This implies that the graph of f (x) = $a^x$  is contained in the first and second quadrants.

For Fig.b
Let 0 < a < 1 and define f : R →R by f(x) = $a^x$ by f (x) = $a^x$  for all x $\in$ R. Sketch the graph of f. Here, f(x) decreases as x increases (since 0 < a < 1) and hence f is a decreasing function. 

The graph of f is the curve shown in Figure b. The curve cuts the y-axis at (0, 1). Also, since a > 0, $a^x$  > 0 for all x $\in$ R. Therefore, the graph of f is contained in the first and second quadrants only.

Simple Problem 1

The half-life of carbon-14 is 5,730 years. If there were 1000 grams of carbon initially, then what is the amount of carbon left after 2000 years? Round your answer to the nearest integer.

Answer:

Using the given data, we can say that carbon-14 is decaying and hence we use the formula of exponential decay.

P = P

e- k t              (*),

Here, P

= initial amount of carbon = 1000 grams.

It is given that the half-life of carbon-14 is 5,730 years. It means

P = P

P/2 = 1000/2 = 500 grams.

Substitute all these values in (*),

500 = 1000 e- k (5730)

Dividing both sides by 1000,

0.5 = e- k (5730)

Taking "ln" on both sides,

ln 0.5 = -5730k

Dividing both sides by -5730,

k = ln 0.5/(-5730) ≈ 0.00012097

We have to find the amount of carbon that is left after 2000 years. Substitute t = 2000 in (*),

P = 1000 e- (0.00012097) (2000) ≈ 785 grams.

So, the amount of carbon left after 1000 years = 785 grams.

Logarithmic Function

Function f(x) = $log_ax$, (x, a > 0) and a $\neq$ 1, is known as logarithmic function. 

Domain of f(x) = (0, ∞) 

and Range of f(x) = R 

Modulus Function → Please visit the following page  CLICK

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