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 fast automobiles. Functions are used in almost all real-world problems which help us with their formulation, interpretation, and solution processes. In mathematics, there are many algebraic functions that make calculations easier.

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This Story also Contains

1. What is Algebraic function?
2. Types of algebraic functions
3. Algebraic Function Graphs
4. Algebraic Function Rules
5. Algebraic Functions vs Transcendental Functions
6. What is Non Algebraic Function?
7. Solved Examples Based On the Algebraic Functions

In this article, we will cover the concepts of the algebraic function class 12. This concept falls under the broader category of sets relation and function. 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 Algebraic function?

An algebraic function is a type of function involving algebraic terms and symbols. Now let us look in detail about what is algebraic function with example.

Algebraic Function Definition

A function $f$ is said to be algebraic if it can be constructed using simple and basic algebraic operations such as addition, subtraction, multiplication, division, and taking roots. They should include only the operations, +, -, ×, ÷, integer and rational exponents. These operations finally result in algebraic functions such as a polynomial function, cubic, quadratic function, linear function, and is also based on the degree of the equations involved.

Algebraic Function Examples

Some examples of algebraic functions are

• - $\mathrm{f}(\mathrm{x})=\mathrm{x}^2-20 \mathrm{x}+89$
• $g(x)=\sqrt{x}$
• $h(x)=(9 x+1) /(6 x-1)$
• $k(x)=x^3$

Identifying Algebraic Functions

If a function includes only the above operations (+, -, ×, ÷, exponents (also roots)), only then can we can conclude that the function is an algebraic function, otherwise not algebraic function.

Domain of Algebraic Functions

The domain of a function is the set of all possible input values for which the function is defined. For algebraic functions, the domain is usually the set of all reals, unless the function has some restrictions.

To find the domain of an algebraic function, we should look for any values that would make the function undefined. For example, in the case of rational functions, the denominator cannot be equal to zero, so we should exclude any values of x that would make the denominator zero. In the case of even radicals (square roots, fourth roots, etc.), the radicand (expression under the radical) must be non-negative, so we should exclude any values of x that would make the radicand negative.

Range of Algebraic Functions

The range of a function is the set of all possible output values. For algebraic functions, the range depends on the function's definition, and it may or may not include all real numbers.

To find the range of an algebraic function, we can use different techniques, such as graphing, analyzing the function's behavior, or solving for the function's outputs. In some cases, the range may be restricted due to the function's behavior. For example, the range of a polynomial function with an even degree will always be non-negative, while the range of a polynomial function with an odd degree will always include all real numbers.

Types of algebraic functions

The types of algebraic function include power function, polynomial and rational functions.

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.
The domain for such type of functions is $R$.
- The range here 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.Few examples include:

• $f(x)=10 x+2$ (linear function)
• $f(x)=x^2-6 x+9$ (quadratic function)
• $f(x)=x^3-30 x+7$ (cubic function)
• $f(x)=x^4-x^2+x-8$ (biquadratic function)
• $f(x)=x^5-67 x+23$ (quintic function)

Power Functions

They can be expressed as $f(x)=k x^a$, where ' $k$ ' and ' $a$ ' are real numbers. The exponent 'a' can take on either integer or rational values. Few examples include:

• $f(x)=x^9$
• $f(x)=x^{-7}$ (reciprocal function)
• $f(x)=(x-8)^{\frac{3}{2}}$

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

Rational Function

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

Few examples include:

• $f(x)=(x-3)(9 x+2)$
• $f(x)=(2 x-7)(x 2-7 x+100)$

Algebraic Function Graphs

The graph of algebraic functions are never same. The equation of the function is responsible for determining the nature of the graph and it changes as the function equation changes. The common process to plot the graph for any function $y = f(x)$ is as follows:

• First we find the x intercept (by setting y = 0).
• Similarly we find the y intercept (by setting x = 0).
• Next, we have to find all asymptotes and plot them.
• The finding of the critical and inflection points is also necessary.
• It is always better to find out some extra points in between every two x-intercepts and in between every two asymptotes for better plotting and analysis since it makes the interpretation easier.
• Finally we plot all these points and join the curves.

Algebraic Function Rules

The algebraic function rules are addition rule, difference rule, product rule, division rule and composition rule.

If $f(x)$ and $g(x)$ are algebraic functions, then:

• $f(x)+g(x)$ (addition) is an algebraic function.
• $f(x)-g(x)$ (subtraction) is an algebraic function.
• $f(x) \cdot g(x)$ (multiplication) is an algebraic function.
• $\frac{f(x)}{g(x)}$ (division, where $g(x) \neq 0$ ) is an algebraic function.
• $ f(g(x))$ (composition) is also an algebraic function.

Important Notes on Algebraic Function

• Algebraic functions include only algebraic operations.
• The algebraic operations included are addition, subtraction, multiplication, division, powers, and roots.
• Any function that involves the term of a log, like trigonometric functions, inverse trigonometric functions, or variable in the exponent is NOT an algebraic function.
• The domain and range of any algebraic function can be found by graphing it on a graphing calculator and seeing the x-values and the y-values respectively that the graph would cover.

Algebraic Functions vs Transcendental Functions

What is Non Algebraic Function?

A non algebraic function is a function with no algebraic terms or symbols. It may be logarithmic, trigonometric or otherbut with no algebraic terms.

Non-Algebraic Function Examples: The list of such functions include trigonometric,logarithmic functions, exponential functions etc. For example , $f(x) = \sin (x), g(x) = logx,$ etc.

Recommended Video Based on Algebraic Functions

Solved Examples Based On the Algebraic Functions

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

Solution:

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

Hence, these are identical functions
Hence, the answer is the option 2.

Example 2: Find the range of the function, $f(x)=\frac{x^2+x+1}{x^2+4 x+3}$.

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

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

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

4) None of these

Solution:

As we learned

Here

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

$\Rightarrow x=-3,-1$

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

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

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

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

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, $f(x)=\frac{2 x-2}{x^2-2 x+3}$,
1) $[-1 / 2,1 / 2]$
2) $[-1,1]$
3) $\left[\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right]$
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 $y=f(x)$
$f(x)=\frac{2 x-2}{x^2-2 x+3}$
Let $y=f(x)=$ i.e $y=\frac{2 x-2}{x^2-2 x+3}$
or $y x^2-2(y+1) x+3 y+2=0$, 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.

$\Delta \geq 0 \Rightarrow(y+1)^2-y(3 y+2) \geq 0 \Rightarrow y^2 \leq \frac{1}{2} \Rightarrow-\frac{1}{\sqrt{2}} \leq y \leq \frac{1}{\sqrt{2}}$

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

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

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