Pre-Calculus - Week 1

Functions and Their Graphs

Functions are the core language of calculus. This week you learn what a function is, how to read function notation, how to find domain and range, and how to recognize and graph the toolkit functions that appear throughout all of mathematics. A solid grip on functions here will accelerate every week that follows.

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 a function from an equation or graph. - Identify and graph the six toolkit functions: linear, quadratic, cubic, absolute value, square root, reciprocal. - Apply vertical and horizontal shifts and reflections to transform graphs. - Determine whether a function is even, odd, or neither. - Find and interpret the average rate of change of a function on an interval.

1. What Is a Function?

A function is a rule that assigns exactly one output to each input. The input variable is usually called x (the independent variable). The output variable is usually called y or f(x) (the dependent variable). The vertical line test: if any vertical line crosses a graph more than once, the graph does not represent a function. Example: y = x^2 is a function. A circle x^2 + y^2 = 25 is NOT a function (a vertical line through x = 3 hits two points: y = 4 and y = -4).

2. Function Notation

f(x) is read 'f of x'. It means the output of function f when the input is x. Example 1: If f(x) = 3x - 1, find f(4). f(4) = 3(4) - 1 = 12 - 1 = 11 Example 2: If f(x) = x^2 + 2x, find f(-3). f(-3) = (-3)^2 + 2(-3) = 9 - 6 = 3 Example 3: Find f(a + 1) for f(x) = 2x - 5. f(a + 1) = 2(a + 1) - 5 = 2a + 2 - 5 = 2a - 3 Note: f(x) is NOT f times x. It is a name for the output.

3. Domain and Range

The domain is the set of all valid inputs (x-values). The range is the set of all possible outputs (y-values). Finding domain algebraically — watch for: 1. Division by zero: exclude values that make the denominator zero. 2. Even roots of negatives: exclude values that make the radicand negative. Example 1: f(x) = 1/(x - 3). Domain: x cannot equal 3. Domain: all reals except 3. Example 2: g(x) = sqrt(2x - 6). Need 2x - 6 >= 0, so x >= 3. Domain: [3, infinity). Example 3: h(x) = x^2 - 4. No restrictions. Domain: all real numbers. Range: [-4, infinity).

4. Toolkit Functions

These six functions appear everywhere in mathematics: 1. Linear: f(x) = x. Straight line through origin, slope 1. 2. Quadratic: f(x) = x^2. U-shaped parabola, vertex at origin. 3. Cubic: f(x) = x^3. S-shaped curve through origin. 4. Absolute Value: f(x) = |x|. V-shape, vertex at origin. 5. Square Root: f(x) = sqrt(x). Domain [0, inf), starts at origin, curves right. 6. Reciprocal: f(x) = 1/x. Two branches, undefined at x = 0, asymptotes at x = 0 and y = 0. Know these shapes cold. Every transformation in precalculus is built on these six.

5. Vertical and Horizontal Shifts

Starting from a base function f(x): Vertical shifts: - f(x) + k shifts the graph UP k units. - f(x) - k shifts the graph DOWN k units. Horizontal shifts (counterintuitive — the sign flips): - f(x - h) shifts the graph RIGHT h units. - f(x + h) shifts the graph LEFT h units. Example 1: g(x) = x^2 + 3 is the parabola shifted up 3. Example 2: g(x) = (x - 2)^2 is the parabola shifted right 2. Example 3: g(x) = (x + 1)^2 - 4 is the parabola shifted left 1 and down 4. Vertex at (-1, -4).

6. Reflections and Vertical Stretches

Reflections: - -f(x): reflect across the x-axis (flip upside down). - f(-x): reflect across the y-axis (flip left-right). Vertical stretches/compressions: - a*f(x) with |a| > 1: stretch away from x-axis. - a*f(x) with 0 < |a| < 1: compress toward x-axis. Example 1: g(x) = -x^2. Parabola opens downward. Example 2: g(x) = 2*sqrt(x). Square root graph stretched vertically by factor 2. Example 3: g(x) = -|x - 3| + 2. Absolute value reflected (opens down), shifted right 3 and up 2. Vertex at (3, 2).

7. Even and Odd Functions

A function is even if f(-x) = f(x) for all x. Even functions are symmetric about the y-axis. Examples: f(x) = x^2, f(x) = |x|, f(x) = x^4 - 3x^2. A function is odd if f(-x) = -f(x) for all x. Odd functions are symmetric about the origin. Examples: f(x) = x, f(x) = x^3, f(x) = 1/x. Test algebraically: 1. Find f(-x). 2. If f(-x) = f(x): even. 3. If f(-x) = -f(x): odd. 4. Otherwise: neither. Example: f(x) = x^3 - x. f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x). ODD.

8. Average Rate of Change

The average rate of change of f on [a, b] is: ARC = (f(b) - f(a)) / (b - a) This is the slope of the secant line connecting (a, f(a)) to (b, f(b)). Example: Find the average rate of change of f(x) = x^2 on [1, 4]. f(1) = 1, f(4) = 16. ARC = (16 - 1)/(4 - 1) = 15/3 = 5. This means on average, f increases 5 units for each 1 unit increase in x on [1, 4]. In calculus, we will let b approach a, shrinking the interval to a single point. That limit is the derivative.

9. Piecewise Functions

A piecewise function uses different formulas on different parts of its domain. Example: f(x) = { x + 2 if x < 0 { x^2 if x >= 0 To evaluate: use the formula whose condition matches. f(-3) = -3 + 2 = -1 (since -3 < 0) f(0) = 0^2 = 0 (since 0 >= 0) f(4) = 4^2 = 16 To find the domain: the full domain is all real numbers (both pieces together cover all x). To find the range: piece 1 gives outputs less than 2; piece 2 gives outputs >= 0. Range: (-inf, 2) union [0, inf) = all reals.

10. Combining Functions

Given two functions f and g: (f + g)(x) = f(x) + g(x) (f - g)(x) = f(x) - g(x) (f * g)(x) = f(x) * g(x) (f/g)(x) = f(x)/g(x), where g(x) is not 0 Composition: (f of g)(x) = f(g(x)). Apply g first, then f to the result. Example: f(x) = 2x + 1, g(x) = x^2. f(g(x)) = f(x^2) = 2x^2 + 1. g(f(x)) = g(2x+1) = (2x+1)^2 = 4x^2 + 4x + 1. Note: composition is generally NOT commutative. f(g(x)) usually differs from g(f(x)).

11. Common Mistakes

- Confusing f(x + 2) with f(x) + 2. These are different: one is a horizontal shift, the other a vertical shift. - Writing the domain as a list of excluded points without using proper interval notation. - Forgetting to check both division and square roots when finding domain. - Thinking the horizontal shift direction matches the sign: f(x - 3) shifts RIGHT (not left). - Evaluating composition in the wrong order: f(g(x)) means apply g first.