# 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**

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

**Square Root Function**

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].

*For Fig.b*

**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

**Contents:**

**Ordered Pair****Cartesian Product****Relations****Different Types of Relations****Composition of Relations****Functions or Mappings****Classification of Functions****Modulus Function****Greatest Integer Function****Inverse Function****Algebra of Real Functions****Composition of Functions**

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