Algebraic Function  

An algebraic function is a function defined by algebraic expressions involving variables, constants, and basic operations like addition, subtraction, multiplication, division, and roots. Examples include polynomials, rational, and radical functions. These functions form the core of algebra and calculus, widely used in science, engineering, and economics. Understanding their types, graphs, and calculus concepts like limits and derivatives is essential for solving real-world problems and advanced mathematical studies.

1.0Algebraic Function Meaning

An algebraic function is a function that can be formed using a finite number of algebraic operations (addition, subtraction, multiplication, division, and taking roots) on the variable x. These functions include polynomials, rational functions, and radical functions.

Example:

$\begin{array}{rl}& f(x)=3x^2+2x-5\\ & g(x)=\frac{x^2+1}{x-3}\\ & h(x)=\sqrt{x+4}\end{array}$

All the above functions are algebraic functions.

2.0Types of Algebraic Functions

Algebraic functions can be classified based on their structure:

1. Polynomial Functions: Involve terms like $x^n$, where n is a non-negative integer.
   Example: $f(x)=x^3+2x+1$
2. Rational Functions: Ratio of two polynomials.

Example: $f(x)=\frac{x^2-1}{x+3}$

3. Radical Functions: Involve roots of polynomials.

Example: $f(x)=\sqrt{x^2+1}$

4. Linear Functions: First-degree polynomials.

Example: $f(x)=2x+5$

5. Quadratic Functions: Second-degree polynomials.

Example: $f(x)=x^2-4x+4$

3.0Algebraic Function Graph

The algebraic function graph varies depending on the type of function:

• Linear function: A straight line.
• Quadratic function: A parabola.
• Cubic function: A curve with one or two turns.
• Rational function: May have vertical and horizontal asymptotes.
• Radical function: Usually starts from a point and curves gradually.

Graph Tip:

To sketch the graph:

1. Identify the domain.
2. Calculate key points (like x- and y-intercepts).
3. Understand asymptotes (if any).
4. Plot and connect smoothly.

4.0Algebraic Function Formula

The general algebraic function formula depends on the type, but here are some examples:

• Polynomial:$f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_0$
• Rational:$f(x)=\frac{P(x)}{Q(x)}$
• Radical:$f(x)=\sqrt[n]{P(x)}$

5.0Limit of Algebraic Function

The limit of an algebraic function is the value the function approaches as the input approaches a specific point.

Example:

${\lim }_{x\to 2}\frac{x^2-4}{x-2}$

Factor and simplify:

$={\lim }_{x\to 2}\frac{(x-2)(x+2)}{x-2}={\lim }_{x\to 2}(x+2)=4$

Limits help determine continuity and prepare the ground for calculus.

6.0Differentiation of Algebraic Functions

Differentiation of algebraic functions involves applying the rules of calculus (like power rule, product rule, quotient rule) to find the derivatives of algebraic function expressions.

Basic Derivatives:

$\begin{array}{rl}& \frac{d}{dx}\left[x^n\right]=n{x}^{n-1}\\ & \frac{d}{dx}[a]=0\\ & \frac{d}{dx}[x]=1\end{array}$

Example:

$\begin{array}{rl}& f(x)=3x^3-5x^2+2x+7\\ & f^{\prime }(x)=9x^2-10x+2\end{array}$

7.0Applications of Algebraic Function

Algebraic functions are applied in a wide range of disciplines:

• Physics: Describing motion, forces, and energy.
• Economics: Modeling supply-demand curves, cost functions.
• Engineering: Circuit analysis, material stress models.
• Computer Graphics: Rendering curves and animations.
• Biology: Modeling population growth or decay.

8.0Solved Examples on Algebraic Functions

Example 1: Find the domain of $f(x)=\frac{x+1}{\sqrt{x-3}}$

Solution:
The denominator contains a square root → must be > 0 (not just ≥ 0 because it’s in the denominator).
So,

$x-3>0\Rightarrow x>3$

Domain: ($3,\infty$)

Example 2: Simplify $f(x)=\frac{x^2-9}{x-3}$  and find f(4)

Solution:
First factor the numerator:

$f(x)=\frac{(x-3)(x+3)}{x-3}$

Cancel common factor ($forx\ne 3$):

f(x) = x + 3 

Now,

f(4) = 4 + 3 = 7 

Answer: f(4) = 7

Example 3: If $f(x)=\sqrt{4-x^2}$, find its domain

Solution:
Expression under square root must be ≥ 0:

$\begin{array}{rl}& 4-x^2\ge 0\\ & \Rightarrow x^2\le 4\\ & \Rightarrow -2\le x\le 2\end{array}$

 Domain: [-2, 2]

Example 4: Find the inverse of $f(x)=\frac{2x-1}{x+3}$

Solution:
Let

$y=\frac{2x-1}{x+3}$

Swap x and y to get inverse:

$x=\frac{2y-1}{y+3}$

Cross-multiply:

$\begin{array}{rl}& x(y+3)=2y-1\\ & \Rightarrow xy+3x=2y-1\\ & \Rightarrow xy-2y=-3x-1\\ & \Rightarrow y(x-2)=-3x-1\\ & \Rightarrow y=\frac{-3x-1}{x-2}\end{array}$ 

Inverse: $f^{-1}(x)=\frac{-3x-1}{x-2}$

Example 5: $Letf(x)=x^2-2x+3$. Find the minimum value of the function.

Solution:
This is a quadratic function $f(x)=a{x}^2+bx+c$ with a = 1 > 0, so it opens upwards.
Minimum value occurs at:

$\begin{array}{rl}& x=-\frac{b}{2a}=-\frac{-2}{2\cdot 1}=1\\ & f(1)=1^2-2\cdot 1+3=1-2+3=2\end{array}$

Minimum value: 2 at x = 1

Example 6. What is an algebraic function in simple terms?

Ans: An algebraic function is a mathematical expression formed using basic algebraic operations—addition, subtraction, multiplication, division, and roots—on a variable (like x). For example, $f(x)=x^2+3x-5$is an algebraic function.

Example 7. How are algebraic functions different from transcendental functions?

Ans: Algebraic functions are built from algebraic operations, while transcendental functions involve non-algebraic operations like exponential, logarithmic, and trigonometric functions. For example:

• Algebraic: $\sqrt{x^2+1}$
• Transcendental: $e^x,\log x,\sin x$

Example 8. What is the formula for an algebraic function?

Ans: There is no single formula, but common forms include:

• Polynomial: $f(x)=a{x}^n+b{x}^{n-1}+\dots +k$
• Rational: $f(x)=\frac{P(x)}{Q(x)}$
• Radical: $f(x)=\sqrt[n]{P(x)}$

Example 9. How do you differentiate an algebraic function?

Ans: Use standard rules of differentiation:

• Power Rule: $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$
• Sum Rule, Product Rule, and Quotient Rule are also used for more complex algebraic expressions.

9.0Algebraic Function Practice Problems

Q1. Find the limit: ​​${\lim }_{x\to 1}\frac{x^2-1}{x-1}$

Q2. Differentiate: $f(x)=\frac{2x^2+3x}{x-1}$

Q3. Sketch the graph of $f(x)=x^2-4$

Q4. Classify the function: $f(x)=\sqrt{x+2}$

Q5. Find the domain of: $f(x)=\frac{1}{x^2-9}$