📈College Algebra Unit 3 – Functions

What Are Functions?

• Functions are mathematical rules that assign a unique output value to each input value
• Denoted as $f(x)$, where $x$ is the input and $f(x)$ is the corresponding output
• Can be represented using equations, graphs, or tables
• The set of all possible input values is called the domain, while the set of all possible output values is called the range
• Functions can be one-to-one, meaning each output value corresponds to exactly one input value
• Many real-world relationships can be modeled using functions (temperature conversion)
• Functions can be continuous, meaning there are no breaks or gaps in the graph, or discontinuous

Types of Functions

• Linear functions have the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept
  • Graphs of linear functions are straight lines ($f(x) = 2x + 1$)
• Quadratic functions have the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$
  • Graphs of quadratic functions are parabolas ($f(x) = x^2 - 4x + 3$)
• 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 $x > 0$
  • Logarithmic functions are the inverses of exponential functions
• Trigonometric functions include sine, cosine, and tangent, which relate angles to ratios of side lengths in right triangles
• Piecewise functions are defined by different equations over different intervals of the domain (absolute value function)

Function Notation and Terminology

• $f(x)$ is read as "f of x" and represents the output value when the input is $x$
• $f(a)$ represents the output value when the input is $a$
• The variable $x$ is often used to represent the input, but other variables can be used (t for time)
• The term "evaluate" means to find the output value for a given input
  • To evaluate $f(3)$ for $f(x) = 2x + 1$, substitute $3$ for $x$: $f(3) = 2(3) + 1 = 7$
• The vertical line test determines if a graph represents a function: if any vertical line intersects the graph more than once, it is not a function
• A function's domain can be restricted due to real-world constraints or to avoid undefined values (square root function)

Graphing Functions

• To graph a function, create a table of input and output values, plot the points, and connect them with a smooth curve
• The x-intercepts of a function are the points where the graph crosses the x-axis ($y = 0$)
• The y-intercept is the point where the graph crosses the y-axis ($x = 0$)
• Symmetry can be used to graph functions more efficiently
  • Even functions are symmetric about the y-axis: $f(-x) = f(x)$ (absolute value function)
  • Odd functions are symmetric about the origin: $f(-x) = -f(x)$ (cubic function)
• Transformations can be applied to function graphs:
  • Vertical shifts: $f(x) + k$ shifts the graph up by $k$ units, $f(x) - k$ shifts it down
  • Horizontal shifts: $f(x - h)$ shifts the graph right by $h$ units, $f(x + h)$ shifts it left
  • Reflections: $-f(x)$ reflects the graph over the x-axis, $f(-x)$ reflects it over the y-axis

Function Operations and Composition

• Functions can be added, subtracted, multiplied, and divided pointwise
  • $(f + g)(x) = f(x) + g(x)$, $(f - g)(x) = f(x) - g(x)$, $(f \cdot g)(x) = f(x) \cdot g(x)$, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ for $g(x) \neq 0$
• Function composition involves substituting one function into another
  • $(f \circ g)(x) = f(g(x))$, read as "f composed with g of x"
  • To evaluate $(f \circ g)(a)$, first evaluate $g(a)$, then use that result as the input for $f$
• Function composition is not always commutative: $(f \circ g)(x)$ may not equal $(g \circ f)(x)$
• Decomposing a composite function into its constituent functions can help simplify complex expressions

Inverse Functions

• The inverse of a function "undoes" the original function
  • If $f(a) = b$, then $f^{-1}(b) = a$, where $f^{-1}$ is the inverse of $f$
• A function and its inverse are reflections of each other over the line $y = x$
• To find the inverse of a function algebraically:
  1. Replace $f(x)$ with $y$
  2. Swap $x$ and $y$
  3. Solve for $y$
  4. Replace $y$ with $f^{-1}(x)$
• Not all functions have inverses; a function must be one-to-one (pass the horizontal line test) to have an inverse
• The domain of a function becomes the range of its inverse, and vice versa

Real-World Applications

• Functions can model relationships between variables in various fields (physics, economics, biology)
• Linear functions can represent constant rates of change (distance traveled over time at a constant speed)
• Quadratic functions can model projectile motion, with height as a function of time
• Exponential functions can describe population growth or radioactive decay
• Logarithmic functions can model sound intensity (decibels) or earthquake magnitude (Richter scale)
• Trigonometric functions are used in wave modeling (sound waves, ocean tides) and periodic phenomena (hours of daylight throughout the year)

Common Mistakes and How to Avoid Them

• Forgetting to use parentheses when substituting values into functions
  • Remember PEMDAS: evaluate what's inside the parentheses first
• Confusing function notation with multiplication
  • $f(x)$ does not mean $f$ times $x$; it represents the output of the function $f$ when the input is $x$
• Misinterpreting the order of operations in function composition
  • Work from the inside out: first evaluate the inner function, then use that result as the input for the outer function
• Attempting to find the inverse of a non-one-to-one function
  • Check that the function passes the horizontal line test before finding its inverse
• Incorrectly graphing transformations of functions
  • Pay attention to the sign of the constants: negative signs result in reflections or shifts in the opposite direction
• Misidentifying the domain and range of a function
  • Consider any restrictions on the input values and the resulting limitations on the output values