A polynomial function has the form:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$

where $a_n, a_{n-1}, \ldots, a_0$ are constants, $n$ is a non-negative integer, and $a_n \neq 0$. The highest power of $x$ is called the degree of the polynomial.

Classification by Degree

• Degree 0: Constant function (e.g., $f(x) = 5$)
• Degree 1: Linear function (e.g., $f(x) = 3x + 2$)
• Degree 2: Quadratic function (e.g., $f(x) = x^2 + 4x + 2$)
• Degree 3: Cubic function (e.g., $f(x) = x^3 - 4x$)
• Degree 4: Quartic function (e.g., $f(x) = x^4 - 4x^3 + 2x^2 + 3x - 1$)

Worked Example: Graphing a Quadratic Function

Example: Draw the graph of $f(x) = x^2 + 4x + 2$

Solution:

1. Create a table of values:

2. Plot the points (-4, 2), (-3, -1), (-2, -2), (-1, -1), (0, 2) on the xy-plane.

3. Connect the points with a smooth curve. The graph is a parabola opening upward (since $a = 1 > 0$). The vertex is at $x = -\frac{b}{2a} = -\frac{4}{2} = -2$, giving the minimum value $f(-2) = -2$.

Domain: All real numbers, $\{x: x \in \mathbb{R}\}$

Range: $\{y: y \geq -2\}$

A rational function has the form:

$f(x) = \frac{p(x)}{q(x)}$

where $p(x)$ and $q(x)$ are polynomial functions and $q(x) \neq 0$.

Key Features

(a) Vertical Asymptotes

Lines where the function is undefined. Found by setting the denominator equal to zero:

$q(x) = 0$

(b) Horizontal Asymptotes

• If degree of numerator equals degree of denominator: $y = \frac{\text{leading coefficient of } p(x)}{\text{leading coefficient of } q(x)}$
• If degree of numerator < degree of denominator: $y = 0$
• If degree of numerator > degree of denominator: no horizontal asymptote

(c) Oblique Asymptotes

When degree of numerator = degree of denominator + 1. Found by polynomial long division. The quotient (ignoring the remainder) is the oblique asymptote.

(d) Holes

occur when both numerator and denominator share a common factor $(x - a)$. The hole is at the point $(a, f(a))$ after simplifying.

Worked Example: Finding Asymptotes and Holes

Example: Find the holes and asymptotes of $f(x) = \frac{2x^2 - 4x}{x^2 + x - 6}$

Solution:

1. Factor numerator and denominator: $f(x) = \frac{2x(x - 2)}{(x + 3)(x - 2)}$

2. The common factor $(x - 2)$ indicates a hole at $x = 2$.

3. Simplify: $f(x) = \frac{2x}{x + 3}$ (for $x \neq 2$)

4. Hole: Substitute $x = 2$ into simplified form: $f(2) = \frac{4}{5}$, so the hole is at $(2, \frac{4}{5})$

5. Vertical asymptote: Set denominator $x + 3 = 0$, so $x = -3$

6. Horizontal asymptote: Both numerator and denominator have degree 1, so $y = \frac{2}{1} = 2$

A composite function is formed by substituting one function into another. If $f(x)$ and $g(x)$ are functions, then:

• $(f \circ g)(x) = f(g(x))$ — apply $g$ first, then $f$
• $(g \circ f)(x) = g(f(x))$ — apply $f$ first, then $g$

The inner function is applied first, and its output becomes the input for the outer function.

Worked Example: Forming Composite Functions

Example: Given $g(x) = 2x - 2$ and $f(x) = x^2 + 5$, find: (a) $(g \circ f)(x)$ (b) $(f \circ g)(x)$

Solution:

(a) $(g \circ f)(x) = g(f(x)) = g(x^2 + 5) = 2(x^2 + 5) - 2 = 2x^2 + 10 - 2 = 2x^2 + 8$

(b) $(f \circ g)(x) = f(g(x)) = f(2x - 2) = (2x - 2)^2 + 5 = 4x^2 - 8x + 4 + 5 = 4x^2 - 8x + 9$

Worked Example: Evaluating Composite Functions

Example: Given $f(x) = 6x + 2$ and $g(x) = x - 2$, find $f[g(6)]$

Solution:

1. First find $f[g(x)] = f(x - 2) = 6(x - 2) + 2 = 6x - 12 + 2 = 6x - 10$

2. Then $f[g(6)] = 6(6) - 10 = 36 - 10 = 26$

An exponential function has the form:

$f(x) = a^{x} \quad \text{or} \quad f(x) = a^{g(x)}$

where the base $a > 0$, $a \neq 1$, and $x$ is any real number.

Properties

• Domain: All real numbers, $\{x: x \in \mathbb{R}\}$
• Range: $y > 0$ (positive values only)
• If $a > 1$: function is increasing
• If $0 < a < 1$: function is decreasing

Special Exponential: The Natural Exponential

The function $f(x) = e^x$ (where $e \approx 2.718$) is called the natural exponential function, important in calculus and modelling natural growth/decay.

Worked Example: Graphing an Exponential Function

Example: Draw the graph of $g(x) = 2^x$

Solution:

1. Create a table of values:

2. Plot the points and connect with a smooth curve.

3. The graph approaches the x-axis as $x \to -\infty$ (horizontal asymptote at $y = 0$) and rises rapidly as $x \to \infty$.

A logarithmic function is the inverse of an exponential function:

$y = \log_a x \quad \text{means} \quad x = a^y$

where $a > 0$ and $a \neq 1$, and $x > 0$.

Properties

• Domain: $x > 0$
• Range: All real numbers, $\{y: y \in \mathbb{R}\}$
• If $a > 1$: logarithmic function is increasing
• If $0 < a < 1$: logarithmic function is decreasing

Special Logarithms

• $\log_{10} x$ is common logarithm (written as $\log x$)
• $\log_e x$ is natural logarithm (written as $\ln x$ or $\log_e x$)

Worked Example: Graphing a Logarithmic Function

Example: Draw the graph of $f(x) = \log_2 x$ and state its domain and range.

Solution:

1. Create a table of values:

2. Plot the points and connect with a smooth curve.

3. The graph approaches the y-axis as $x \to 0^+$ (vertical asymptote at $x = 0$) and rises slowly as $x \to \infty$.

Domain: $\{x: x > 0\}$

Range: $\{y: y \in \mathbb{R}\}$

In Tanzania, these function concepts are used in many practical situations. For example, when a mobile phone trader wants to model the relationship between the number of airtime units sold and revenue earned, exponential functions can describe growth in sales over time, while logarithmic functions help analyze data that grows quickly at first then levels off. A shopkeeper in Dar es Salaam might use polynomial functions to calculate profit based on different pricing strategies, or use composite functions to determine the final price after applying multiple discounts. Understanding these functions enables better decision-making in small businesses, farming (predicting crop yields), and financial planning (calculating interest on savings or loans at banks).