Algebraic function - Practice Questions & MCQ

Algebraic function:

A function f is said to be algebraic if it can be constructed using algebraic operations such as addition, subtraction, multiplication, division and taking roots.  E.g.  

$f(x)=\sqrt{1+x}$

Monomial function 

A function of the form $\mathrm{y}=\mathrm{ax}{ }^{\mathrm{n}}$, where $\mathbf{a}$ is constant and $\mathbf{n}$ is a non-negative integer, is called a monomial function.

$
\text { E.g } y=x^2, y=2 x, y=-x, e t c
$
 

             y = x²                                                                    y = 2x
Polynomial function

A real-valued function $f: R \rightarrow R$ defined by $y=f(x)=a_0+a_1 x+a_2 x^2 \ldots+a_n x^n$, where $n \in N$, and $a_0, a_1, a_2$ $\ldots a_n \in R$, for each $x \in R$, is called Polynomial functions.

The highest power of $x$ is called the Degree of this polynomial.
Domain for such functions is R.
The range depends on the degree of the polynomial. If the degree is odd, then the range is R, but it does not equal $R$ if the degree is even.

Identity function

Let $R$ be the set of real numbers. Define the real-valued function $f: R \rightarrow R$ by $y=f(x)=x$ for each $x \in R$.
Such a function is called the identity function. It is denoted by $I_A$. Here the domain and range of function are R. The graph is a straight line.

                                     y = x

Constant function

The function $f: R \rightarrow R$ by $y=f(x)=c, x \in R$ where $c$ is a constant and each $x \in R$.
Here, the domain of f is R and its range is $\{c\}$.
The graph is a line parallel to the $x$-axis. For example, if $f(x)=4$ for each $x \in R$, then its graph will be a line as shown in Fig

As from the above figure, we can see that the blue line is $y=4$
Green line is $\mathrm{y}=2$ and purple line is $\mathrm{y}=-2$
Rational Function

$f(x)=\frac{\rho(x)}{q(x)}$, Where $\rho(x)$ and $q(x)_{\text {polynomials in } \mathrm{x} \text {. }}$
Domain of this function is $R-\{x: q(x)=0\}$
Range depends on the function.