Differential Calculus Unit 1 Review: Functions and Graphs

Key Concepts and Definitions

• Functions map each input value (usually denoted as $x$) to a unique output value (usually denoted as $y$)
• The vertical line test determines if a relation is a function
  • If any vertical line intersects the graph more than once, it is not a function
• Domain refers to the set of all possible input values ($x$-values) for a function
• Range refers to the set of all possible output values ($y$-values) for a function
• One-to-one functions have a unique output value for each input value and pass the horizontal line test
• Even functions are symmetric about the $y$-axis ($f(-x) = f(x)$)
• Odd functions are symmetric about the origin ($f(-x) = -f(x)$)

Function Basics and Types

• Functions can be represented using equations, graphs, tables, or verbal descriptions
• Linear functions have a constant rate of change and are represented by the equation $y = mx + b$
  • $m$ represents the slope and $b$ represents the $y$-intercept
• Quadratic functions are represented by the equation $y = ax^2 + bx + c$
  • They have a parabolic shape and can open upward ($a > 0$) or downward ($a < 0$)
• Polynomial functions are the sum of terms with non-negative integer exponents (linear, quadratic, cubic, etc.)
• Rational functions are the quotient of two polynomial functions ($\frac{P(x)}{Q(x)}$)
• Exponential functions have a constant base raised to a variable power ($y = a^x$)
• Logarithmic functions are the inverse of exponential functions ($y = \log_a(x)$)
• Trigonometric functions (sine, cosine, tangent) relate angles to the sides of a right triangle

Graphing Techniques

• To graph a function by hand, create a table of input and output values, plot the points, and connect them smoothly
• Identify key features of a graph, such as intercepts, symmetry, and asymptotes
• $x$-intercepts occur when $y = 0$ and can be found by factoring or setting the function equal to zero
• $y$-intercepts occur when $x = 0$ and can be found by evaluating the function at $x = 0$
• Horizontal asymptotes describe the behavior of a function as $x$ approaches positive or negative infinity
  • For rational functions, compare the degrees of the numerator and denominator to determine horizontal asymptotes
• Vertical asymptotes occur when the denominator of a rational function equals zero
• Sketch the general shape of a function by analyzing its equation and key features

Domain and Range

• The domain of a function can be determined by identifying any values of $x$ that would make the function undefined
  • For rational functions, exclude values that make the denominator equal to zero
  • For functions with even roots (e.g., $\sqrt{x}$), exclude values that would result in a negative value under the radical
• The range of a function can be determined by analyzing its graph or equation
  • Identify the lowest and highest points on the graph (if they exist)
  • Consider any horizontal asymptotes or gaps in the graph
• Functions can have restricted domains due to real-world limitations (e.g., time cannot be negative)
• To find the range of a function algebraically, set $y$ equal to the function and solve for $x$
  • The resulting inequality for $x$ describes the range of the function

Function Transformations

• Transformations alter the graph of a function without changing its basic shape
• Vertical shifts move the graph up or down and are represented by adding or subtracting a constant to the function ($y = f(x) \pm k$)
• Horizontal shifts move the graph left or right and are represented by adding or subtracting a constant to the input ($y = f(x \pm h)$)
  • A positive $h$ shifts the graph left, while a negative $h$ shifts the graph right
• Vertical stretches or compressions multiply the function by a constant ($y = af(x)$)
  • $|a| > 1$ stretches the graph vertically, while $0 < |a| < 1$ compresses the graph vertically
• Horizontal stretches or compressions multiply the input by a constant ($y = f(bx)$)
  • $|b| > 1$ compresses the graph horizontally, while $0 < |b| < 1$ stretches the graph horizontally
• Reflections flip the graph across the $x$-axis ($y = -f(x)$) or the $y$-axis ($y = f(-x)$)

Composition and Inverse Functions

• Function composition combines two functions by using the output of one function as the input of another ($f \circ g(x) = f(g(x))$)
• To find $(f \circ g)(x)$, first evaluate $g(x)$, then use the result as the input for $f(x)$
• The domain of a composite function is the set of all $x$ values for which both $g(x)$ and $f(g(x))$ are defined
• Inverse functions "undo" each other ($f(f^{-1}(x)) = f^{-1}(f(x)) = x$)
  • The graph of an inverse function is a reflection of the original function across the line $y = x$
• To find the inverse of a function algebraically, swap $x$ and $y$, then solve for $y$
• The domain of a function becomes the range of its inverse, and vice versa
• Not all functions have inverses; a function must be one-to-one (pass the horizontal line test) to have an inverse

Important Function Families

• Absolute value functions ($y = |x|$) have a V-shaped graph with the vertex at the origin
• Square root functions ($y = \sqrt{x}$) have a domain of $x \geq 0$ and a range of $y \geq 0$
• Cubic functions ($y = x^3$) have an S-shaped graph and are odd functions
• Exponential functions ($y = a^x$) have a horizontal asymptote at $y = 0$ when $0 < a < 1$ and increase rapidly when $a > 1$
• Logarithmic functions ($y = \log_a(x)$) have a vertical asymptote at $x = 0$ and a domain of $x > 0$
• Trigonometric functions (sine, cosine, tangent) are periodic and have specific ranges
  • Sine and cosine have a range of [-1, 1], while tangent has a range of $(-\infty, \infty)$

Real-World Applications

• Linear functions can model situations with a constant rate of change (e.g., cost per item, speed)
• Quadratic functions can model projectile motion, profit optimization, and area problems
• Exponential functions can model population growth, compound interest, and radioactive decay
• Logarithmic functions can model sound intensity (decibels), earthquake magnitude (Richter scale), and pH levels
• Trigonometric functions can model periodic phenomena such as sound waves, tides, and electrical signals
• Rational functions can model inverse proportional relationships (e.g., resistance in electrical circuits)
• Piecewise functions can model situations with different rules for different intervals (e.g., tax brackets, shipping rates)