College Algebra - Week 4

Functions and Function Notation

Functions are the central organizing idea of all of mathematics beyond arithmetic. This week you learn what makes a relation a function, how to use and interpret function notation f(x), how to determine the domain and range of a function, and how to identify function behavior from graphs and tables. This vocabulary and notation will appear in every math course you take from here forward.

Learning Objectives

- Define a function and explain the vertical line test. - Use function notation f(x) to evaluate and interpret functions. - Find the domain and range of functions expressed as equations, graphs, or tables. - Identify when a function is increasing, decreasing, or constant on an interval. - Evaluate and simplify the difference quotient (f(x+h) - f(x))/h. - Compose two functions and find the domain of a composition. - Determine whether a function is one-to-one and find basic inverse functions.

1. Relations vs. Functions

A relation is any set of ordered pairs (x, y). A function is a special relation where each input (x-value) produces exactly ONE output (y-value). Non-function example: {(1,2), (1,3)} — the input 1 gives two outputs. Function example: {(1,2), (2,3), (3,4)} — each input gives exactly one output. The vertical line test for graphs: if any vertical line crosses a graph more than once, the graph is NOT a function.

2. Function Notation

We write f(x) to mean: 'the output of function f when the input is x.' f(x) is read 'f of x' — it does NOT mean f times x. Example: Let f(x) = 3x^2 - 2x + 1. f(0) = 3(0)^2 - 2(0) + 1 = 1 f(2) = 3(4) - 4 + 1 = 9 f(-1) = 3(1) - 2(-1) + 1 = 3 + 2 + 1 = 6 You can substitute any expression, not just numbers: f(a + 1) = 3(a+1)^2 - 2(a+1) + 1

3. Domain and Range

The domain is the set of all valid inputs (x-values). The range is the set of all resulting outputs (y-values). For most functions in algebra, the domain is all real numbers EXCEPT: - Values that cause division by zero. - Values that make a square root of a negative number. Example 1: f(x) = 1/(x - 3) Domain: x cannot equal 3. In interval notation: (-inf, 3) U (3, inf). Example 2: f(x) = sqrt(x + 5) Domain: x + 5 >= 0, so x >= -5. Interval: [-5, inf).

4. Domain Restrictions: Practice

Example 3: f(x) = sqrt(2x - 6) Require: 2x - 6 >= 0 → 2x >= 6 → x >= 3. Domain: [3, inf). Example 4: g(x) = 1/sqrt(x + 4) Require: x + 4 > 0 (need x+4 positive, not just non-negative, because it is in the denominator). x > -4. Domain: (-4, inf). Example 5: h(x) = (x + 2)/((x - 1)(x + 3)) Domain excludes x = 1 and x = -3. Interval: (-inf, -3) U (-3, 1) U (1, inf).

5. Reading Range from Graphs

To find the range from a graph: 1. Look at all y-values that appear on the graph. 2. Think vertically — scan from bottom to top. A parabola opening upward y = x^2: - Domain: all real numbers, (-inf, inf). - Range: y >= 0, i.e., [0, inf). A horizontal line y = 4: - Domain: all real numbers. - Range: {4} (just one value). The square root function y = sqrt(x): - Domain: [0, inf). - Range: [0, inf).

6. Increasing, Decreasing, Constant

A function is increasing on an interval if f(a) < f(b) whenever a < b in that interval. A function is decreasing if f(a) > f(b) whenever a < b. A function is constant if f(a) = f(b) for all a, b in that interval. For f(x) = x^2: - Decreasing on (-inf, 0) - Increasing on (0, inf) - Minimum at x = 0 Note: We describe increasing/decreasing using x-values (the domain interval), not y-values.

7. The Difference Quotient

The difference quotient (f(x+h) - f(x))/h is the foundation of the derivative. Example: Find the difference quotient for f(x) = 2x^2 + 3. Step 1: Find f(x + h). f(x + h) = 2(x + h)^2 + 3 = 2(x^2 + 2xh + h^2) + 3 = 2x^2 + 4xh + 2h^2 + 3 Step 2: Subtract f(x). f(x+h) - f(x) = 2x^2 + 4xh + 2h^2 + 3 - (2x^2 + 3) = 4xh + 2h^2 Step 3: Divide by h. (4xh + 2h^2)/h = 4x + 2h As h approaches 0, the difference quotient approaches 4x — the derivative!

8. Function Composition

The composition (f ∘ g)(x) = f(g(x)) means: apply g first, then apply f to the result. Example: Let f(x) = x^2 + 1 and g(x) = 3x - 2. f(g(x)) = f(3x - 2) = (3x - 2)^2 + 1 = 9x^2 - 12x + 4 + 1 = 9x^2 - 12x + 5 g(f(x)) = g(x^2 + 1) = 3(x^2 + 1) - 2 = 3x^2 + 3 - 2 = 3x^2 + 1 Note: f(g(x)) is NOT the same as g(f(x)) in general.

9. One-to-One Functions and Inverses

A function is one-to-one (1-1) if each y-value is produced by at most one x-value. Horizontal line test: if any horizontal line crosses the graph more than once, the function is NOT one-to-one. Inverse function f^(-1) reverses the roles of x and y. To find the inverse: 1. Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with f^(-1)(x). Example: f(x) = 3x + 2. y = 3x + 2 → x = 3y + 2 → x - 2 = 3y → y = (x - 2)/3. So f^(-1)(x) = (x - 2)/3.

10. Evaluating from Tables and Graphs

From a table: f(3) = the y-value in the row where x = 3. From a graph: f(3) = the y-coordinate at x = 3. Example from a table: x | f(x) 0 | 4 1 | 7 2 | 10 3 | 13 f(2) = 10. What x gives f(x) = 7? Answer: x = 1. This kind of reading is critical in real-world function problems where no formula is given.

11. Piecewise Functions

A piecewise function uses different formulas on different parts of the domain. Example: f(x) = { x + 2 if x < 0; x^2 if x >= 0 } f(-3) = -3 + 2 = -1 (use first piece since -3 < 0) f(0) = 0^2 = 0 (use second piece since 0 >= 0) f(4) = 4^2 = 16 Piecewise functions appear frequently in real applications: tax brackets, shipping rates, phone plans.

12. Common Mistakes

- Writing f(x + 2) = f(x) + f(2). This is almost always wrong. - Confusing domain restrictions: for sqrt(x - 5), require x - 5 >= 0, not x >= 0. - Confusing f^(-1)(x) with 1/f(x). The inverse is not the reciprocal. - For composition: f(g(x)) means apply g FIRST. The notation can be misleading. - Vertical line test answers 'is this a function?' Horizontal line test answers 'is this one-to-one?'