Algebraic Function - Definition, Examples, Types

One of the fundamental ideas in mathematics with a wide range of practical applications is algebraic functions. The methodical use of algebraic functions is necessary for modeling structures such as giant buildings and ultrafast automobiles. Functions are used in almost all real-world issue formulation, interpretation, and solution processes. In mathematics, there are many algebraic functions that make calculations easier.

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In this article, we will cover the concepts of the algebraic function. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

What is an 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.

An algebraic function that can be expressed as a root of an equation or constructed in a finite number of steps from the elementary operations of addition, subtraction, multiplication, division, and exponentiation and the inverse of any function already constructed, The dependence of the function on the independent variables is determined by an algebraic equation. E.g. $f(x)=\sqrt{1+x}$

Types of algebraic function

Polynomial function

A real-valued function $f:R\to R$ defined by $y=f(x)=a_0+a_{1}x+a_2{x}^2\dots +a_n{x}^n$, where $n\in N$, and $a_0,a_1,a_2\dots a_n\in R$, for each $x\in R$, is called Polynomial functions.
The highest power of $x$ is called the Degree of this polynomial.
The 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.

Power Functions

Power functions can be expressed in the form $f(x)=k{x}^a$, where ' $k$ ' and ' $a$ ' are real numbers. The exponent 'a' can take on either integer or rational values. Here are a few examples of power functions.
- $f(x)=x^2$
- $f(x)=x^{-1}$ (reciprocal function)
- $f(x)=(x-2{)}^{1/2}$

The domain of power functions can vary depending on the x-values where the function is defined. Meanwhile, the range of power functions is determined by the y-values covered by the graph of the function.

Rational Function

- Domain of this function is $R-\{x:q(x)=0\}$
- Range depends on the function.

Monomial function

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

$\mathrm{E}.\mathrm{g}y=x^2,y=2x,y=-x,\text{etc}$

y = x² y = 2x

Identity function
Let $R$ be the set of real numbers. Define the real-valued function $f:R\to R$ 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

The function $f:R\to R$ by $y=f(x)=c,x\in R$ where $c$ is a constant and each $x\in R$.
Here 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 $y=2$ and purple line is $y=-2$

Summary

Algebraic functions are versatile and widely used in mathematics due to their straightforward formulation and extensive applications in various fields such as physics, engineering, economics, and more. Understanding their properties and behaviors is essential for solving complex mathematical problems and modeling real-world scenarios.

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Solved Examples Based On the Algebraic Functions:

Example 1: Which of these is an identity function in $x$ ?
1) $f(x)=\sin ({\sin }^{-1}x)$
2) $g(x)=\ln e^x$
3) $f(x)=e^{\ln x}$
4) None of these

Solution:

$\begin{array}{rl}& f(x)=\ln e^x=x\ln e=x\\ & f(x)=x\end{array}$

Hence, these are identical functions
Hence, the answer is the option 2.
Example 2: Find the range of the function, $f(x)=\frac{x^2+x+1}{x^2+4x+3}$.

$\text{1)}y\in [-\mathrm{\infty },\frac{-2-\sqrt{7}}{2}]\cup [\frac{-2+\sqrt{7}}{2},\mathrm{\infty }]$

2) $y\in (-10,10)$
3) $y\in (-8,8)$

4) None of these

Solution:

As we learned

Here

$f(x)=\frac{x^2+x+1}{x^2+4x+3}$ and $\mathrm{f}(\mathrm{x})$ is defined for all $x\in R$ other numbers where $x^2+4x+3=0$

$\Rightarrow x=-3,-1$

Hence the domain of $f(x)=R-(-3,-1)$.

$\frac{x^2+x+1}{x^2+4x+3}=y\Rightarrow x^2(1-y)+x(1-4y)+1-3y=0$

Since x is real, $(1-4y{)}^2-4(1-y)(1-3y)\ge 0\Rightarrow 4y^2+8y-3\ge 0$

$\Rightarrow y\in [-\mathrm{\infty },\frac{-2-\sqrt{7}}{2}]\cup {[\frac{-2+\sqrt{7}}{2},\mathrm{\infty }]}_{\text{which is the required range}}$

Hence, the answer is the option 1.
Example 3: Find the domain of the function, $f(x)=\frac{1}{x+1}$.
1) $R-\{1\}$
2) $R-\{-1\}$
3) $\{-1,1\}$
4) $R$

Solution:

The function is not defined when $x+1=0$ or $x=-1$
Hence the domain is $\mathrm{R}-\{-1\}$
Hence, the answer is the option 2.
Example 4: Find the range of the function, $f(x)=\frac{2x-2}{x^2-2x+3}$,
1) $[-1/2,1/2]$
2) $[-1,1]$
3) $[\frac{-1}{\sqrt{2}},\frac{1}{\sqrt{2}}]$
4) $[-2,2]$

Solution:
As we learned in methods to find the range of the function
For the range of $y=f(x)$, we can first express $x$ as a function of $y:x=g(y)$. Now the domain of $x=g(y)$ is the same as the range of $y=f(x)$
$f(x)=\frac{2x-2}{x^2-2x+3}$
Let $y=f(x)=$ i.e $y=\frac{2x-2}{x^2-2x+3}$
or $y{x}^2-2(y+1)x+3y+2=0$, which is a quadratic in $x$.

For the above quadratic equation to have real roots (real values of $x$ ).

Discriminant should be 0 or positive.

$\mathrm{\Delta }\ge 0\Rightarrow (y+1{)}^2-y(3y+2)\ge 0\Rightarrow y^2\le \frac{1}{2}\Rightarrow -\frac{1}{\sqrt{2}}\le y\le \frac{1}{\sqrt{2}}$

Hence, the range of $f(x)$ is $[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}]$.
Hence, the answer is the option 3.
Solution:

$\begin{array}{rl}& f(x)=\frac{4^x}{4^x+2}\\ & f(1-x)=\frac{\frac{4}{4^x}}{\frac{4}{4x}+2}=\frac{4}{4+2\cdot 4^x}=\frac{2}{2+4^x}\\ & f(x)+f(1-x)=1\end{array}$
Hence, the answer is 1011.

Frequently Asked Questions(FAQ)-

1. What is an algebraic function?

Ans: 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.

2. What is a monomial function?

Ans: A function of the form $y=a{x}^n$, where $\mathbf{a}$ is constant and $\mathbf{n}$ is a non-negative integer, is called a monomial function.
3. What is the degree of polynomial?

Ans: The highest power of a polynomial is called the Degree of this polynomial.
4. What is an identity function?

Ans: The real-valued function $f:R\to R$ by $y=f(x)=x$ for each $x\in R$. Such a function is called the identity function
5. What is a constant function?

Ans: A function whose value is fixed with any domain is called a constant function.