📐Analytic Geometry and Calculus Unit 1 – Precalculus Review: Functions and Graphs

Key Concepts and Definitions

• Functions map input values from the domain to output values in the range
• Function notation $f(x)$ represents the output value of the function $f$ for a given input value $x$
• One-to-one functions have a unique output value for each input value
• Even functions are symmetric about the y-axis, satisfying $f(-x) = f(x)$
• Odd functions are symmetric about the origin, satisfying $f(-x) = -f(x)$
• Piecewise functions are defined by different equations over different intervals of the domain
• Continuous functions have no breaks or gaps in their graphs
  • Discontinuities can be classified as removable, jump, or infinite

Types of Functions

• Linear functions have the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept
• Quadratic functions have the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$
  • The graph of a quadratic function is a parabola
• 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, $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$
• Exponential functions have the form $f(x) = a \cdot b^x$, where $a$ and $b$ are constants, $a \neq 0$, and $b > 0$
• Logarithmic functions have the form $f(x) = \log_b(x)$, where $b$ is the base and $x > 0$
• Trigonometric functions (sine, cosine, tangent) relate angles to the sides of a right triangle

Graphing Techniques

• Plotting points $(x, y)$ on the Cartesian coordinate system
• Identifying x and y intercepts by setting $y = 0$ and $x = 0$, respectively
• Finding the domain and range of a function from its graph
• Determining the intervals where a function is increasing, decreasing, or constant
• Locating local maxima and minima (turning points) on the graph
• Identifying asymptotes (vertical, horizontal, or oblique) of a function
• Sketching the graph of a function using transformations of parent functions
  • Translations, reflections, stretches, and compressions

Domain and Range

• The domain is the set of all possible input values (x-values) for a function
• The range is the set of all possible output values (y-values) for a function
• Determine the domain by considering the function's equation and any restrictions on the input values
  • Avoid division by zero and negative values under even roots
• Determine the range by considering the function's graph or by solving for y in terms of x
• Express the domain and range using interval notation or set-builder notation
• Functions with restricted domains (e.g., square root, logarithm) require special attention

Function Transformations

• Translations shift the graph horizontally or vertically
  • $f(x) + k$ shifts the graph vertically by $k$ units
  • $f(x - h)$ shifts the graph horizontally by $h$ units
• Reflections flip the graph across the x-axis or y-axis
  • $-f(x)$ reflects the graph across the x-axis
  • $f(-x)$ reflects the graph across the y-axis
• Stretches and compressions change the graph's scale
  • $af(x)$ stretches (if $|a| > 1$) or compresses (if $0 < |a| < 1$) the graph vertically by a factor of $|a|$
  • $f(bx)$ compresses (if $|b| > 1$) or stretches (if $0 < |b| < 1$) the graph horizontally by a factor of $\frac{1}{|b|}$

Composition and Inverse Functions

• Function composition combines two functions, $(f \circ g)(x) = f(g(x))$
  • Evaluate the inner function first, then use the result as the input for the outer function
• The inverse function $f^{-1}(x)$ "undoes" the original function $f(x)$
  • If $f(a) = b$, then $f^{-1}(b) = a$
• To find the inverse function, swap $x$ and $y$, then solve for $y$
• A function must be one-to-one to have an inverse function
• The graphs of a function and its inverse are symmetric about the line $y = x$

Real-World Applications

• Linear functions model constant growth or decay (population growth, depreciation)
• Quadratic functions model projectile motion and optimization problems
• Exponential functions model compound interest and exponential growth or decay (bacterial growth, radioactive decay)
• Logarithmic functions model sound intensity (decibels) and earthquake magnitude (Richter scale)
• Trigonometric functions model periodic phenomena (sound waves, tides, seasons)
• Rational functions model inverse proportionality (work rate problems, concentration of solutions)

Common Pitfalls and Tips

• Pay attention to the function's domain and any restrictions on the input values
• Remember the order of operations when evaluating functions (PEMDAS)
• Be careful with negative signs when applying function transformations
• When finding the inverse of a function, check that the original function is one-to-one
• Sketch graphs using key features (intercepts, asymptotes, turning points) before plotting points
• Practice identifying functions from their equations and graphs
• Use technology (graphing calculators, online tools) to verify your work and explore more complex functions