Functions - Definition, Types, Domain Range and Video Lesson - Free Printable

The image you've uploaded is an educational chart from BYJU'S that illustrates different types of mathematical functions, each with its graph and formula. It's a visual guide to help understand the behavior and characteristics of various function types.

Let’s go through each type of function shown in the image, explain what it is, and describe its graph and formula:

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1. Identity Function

- Formula: $ f(a) = b = a $
- Graph: A straight line passing through the origin (0,0) with a slope of 1.
- Explanation: This function maps every input $ a $ to itself. So, if $ a = 3 $, then $ f(3) = 3 $. The graph is linear and passes through points like (1,1), (2,2), etc.

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2. Constant Function

- Formula: $ f(a) = b = 4.5 $
- Graph: A horizontal line at $ y = 4.5 $.
- Explanation: Regardless of the input value $ a $, the output is always 4.5. This means no matter what $ a $ is, $ f(a) = 4.5 $. The graph is flat and parallel to the x-axis.

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3. Polynomial Function

- Formula: Not explicitly given, but the graph looks like a straight line.
- Example: Could be $ f(a) = 2a - 8 $ or any linear polynomial.
- Graph: A straight line with positive slope.
- Explanation: Polynomial functions are expressions involving variables raised to non-negative integer powers. This particular graph appears to be a linear polynomial (degree 1). More complex polynomials can have curves (like quadratics or cubics).

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4. Quadratic Function

- Formula: $ f(a) = b = a^2 - 4 $
- Graph: A parabola opening upwards, vertex at $ (0, -4) $.
- Explanation: Quadratic functions have the form $ f(a) = a^2 + ba + c $. Here, $ f(a) = a^2 - 4 $ has roots at $ a = \pm 2 $ (since $ a^2 - 4 = 0 \Rightarrow a = \pm 2 $). The vertex is at $ (0, -4) $, which is the minimum point.

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5. Cubic Function

- Formula: $ f(a) = a^3 - 5 $
- Graph: S-shaped curve (sigmoid-like), crossing the x-axis at $ a = \sqrt[3]{5} \approx 1.71 $.
- Explanation: Cubic functions have degree 3. They typically have one inflection point and can have up to two turning points. For $ f(a) = a^3 - 5 $, as $ a \to \infty $, $ f(a) \to \infty $, and as $ a \to -\infty $, $ f(a) \to -\infty $. The graph crosses the x-axis where $ a^3 = 5 $.

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6. Rational Function

- Formula: Not explicitly written, but the graph shows asymptotes.
- Typical Example: $ f(a) = \frac{1}{a} $ or $ f(a) = \frac{a}{a-2} $
- Graph: Has vertical and horizontal asymptotes.
- Explanation: Rational functions are ratios of two polynomials. In this case, there is a vertical asymptote at $ a = 0 $ (where denominator becomes zero) and possibly a horizontal asymptote depending on degrees. The graph breaks into two parts around the asymptote.

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7. Modulus Function

- Formula: $ f(a) = b = |a| $
- Graph: V-shaped graph with vertex at the origin (0,0).
- Explanation: The modulus (or absolute value) function returns the non-negative value of $ a $. So:
- If $ a \geq 0 $, then $ |a| = a $
- If $ a < 0 $, then $ |a| = -a $
- Thus, $ f(-3) = 3 $, $ f(3) = 3 $, $ f(0) = 0 $

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Summary Table:

| Function Type | Formula | Key Features |
|---------------------|----------------------|----------------------------------------------|
| Identity | $ f(a) = a $ | Diagonal line through origin |
| Constant | $ f(a) = 4.5 $ | Horizontal line |
| Polynomial | e.g., $ f(a) = 2a - 8 $ | Linear (here), general: sum of powers |
| Quadratic | $ f(a) = a^2 - 4 $ | Parabola, opens upward, vertex at (0,-4) |
| Cubic | $ f(a) = a^3 - 5 $ | S-shaped, one real root near 1.71 |
| Rational | e.g., $ f(a) = \frac{1}{a} $ | Asymptotes, undefined at certain values |
| Modulus | $ f(a) = |a| $ | V-shaped, symmetric about y-axis |

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Final Note:

This chart is excellent for understanding how different algebraic expressions behave visually. Each function type has unique properties:
- Linear/identity: Direct proportion
- Constant: No change in output
- Quadratic: Symmetric U-shape
- Cubic: Can have local max/min
- Rational: May have discontinuities
- Modulus: Reflects negative inputs

These are foundational concepts in algebra and calculus.

Let me know if you'd like to analyze a specific function or solve a related problem!