# Classification of Functions

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.

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

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