1.1 Review of Functions

Functions are the foundation of calculus. They describe how one quantity depends on another, and nearly every concept you'll encounter in this course builds on them. This review covers function notation, domain and range, graphing features, operations, and transformations.

Function Fundamentals

Function notation and evaluation

Function notation $f(x)$ names both the function and its input. The symbol $f(x)$ represents the output value when the input is $x$. To evaluate $f(a)$, substitute $a$ for every $x$ in the function's equation, then simplify.

For example, if $f(x) = 3x^2 - 1$, then $f(2) = 3(2)^2 - 1 = 11$.

Here are the major function types you should recognize:

• Linear functions: $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept
• Quadratic functions: $f(x) = ax^2 + bx + c$, producing a parabola
• Polynomial functions: $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, with non-negative integer exponents
• Rational functions: $f(x) = \frac{P(x)}{Q(x)}$, a ratio of two polynomials
• Exponential functions: $f(x) = a^x$, where $a > 0$ and $a \neq 1$
• Logarithmic functions: $f(x) = \log_a(x)$, the inverse of the exponential, where $a > 0$ and $a \neq 1$
• Trigonometric functions: $\sin(x)$, $\cos(x)$, $\tan(x)$, etc.
• Piecewise functions: defined by different equations over different intervals of the domain

Domain and range identification

The domain is the set of all valid inputs ($x$-values) for a function. Most functions accept all real numbers, but three common restrictions come up repeatedly:

• Rational functions: exclude any $x$ that makes the denominator zero. For $f(x) = \frac{1}{x-3}$, the domain is all real numbers except $x = 3$.
• Square root (and even-root) functions: the expression under the radical must be $\geq 0$. For $f(x) = \sqrt{x - 2}$, the domain is $x \geq 2$.
• Logarithmic functions: the argument must be strictly positive. For $f(x) = \ln(x + 5)$, the domain is $x > -5$.

The range is the set of all possible outputs ($y$-values). Determining range depends on the function type:

• Linear functions (with nonzero slope): range is all real numbers
• Quadratic functions: range depends on whether the parabola opens up ($a > 0$, range $[k, \infty)$) or down ($a < 0$, range $(-\infty, k]$), where $k$ is the vertex's $y$-value
• Exponential functions like $a^x$: range is $(0, \infty)$, always positive
• Logarithmic functions: range is all real numbers
• Trigonometric functions have bounded ranges (e.g., $-1 \leq \sin(x) \leq 1$)

Function graphing and features

When you look at a function's graph, these are the features to identify:

• x-intercepts (zeros): points where the graph crosses the x-axis ($y = 0$)
• y-intercept: the point where the graph crosses the y-axis ($x = 0$); find it by evaluating $f(0)$
• Symmetry: even functions are symmetric across the y-axis; odd functions are symmetric about the origin
• Asymptotes:
  • Vertical asymptotes occur at $x$-values where the function is undefined and the output grows without bound (common in rational functions where the denominator equals zero)
  • Horizontal asymptotes describe the value the function approaches as $x \to \infty$ or $x \to -\infty$
• Intervals of increase/decrease: where the function's output is rising or falling as $x$ moves to the right
• Local maxima/minima: peaks and valleys within a specific interval
• Concavity: whether the curve bends upward (concave up) or downward (concave down)
• Inflection points: where the concavity switches direction

Zeros of functions

Zeros (also called roots) are the $x$-values where $f(x) = 0$. Finding them is a skill you'll use constantly.

Algebraic methods:

1. Factoring: Rewrite the function as a product of simpler expressions, then set each factor equal to zero. For $f(x) = x^2 - 5x + 6 = (x-2)(x-3)$, the zeros are $x = 2$ and $x = 3$.

2. Quadratic formula: When factoring isn't obvious, use $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for any quadratic $ax^2 + bx + c = 0$.

3. Rational root theorem: For higher-degree polynomials, the possible rational zeros are $\pm \frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient. Test candidates by substitution or synthetic division.

Graphical method: Look for x-intercepts on the graph. This gives approximate zeros, which you can then verify algebraically.

Function Representations and Operations

Function representations

Functions can be expressed in three main ways, and you should be comfortable moving between them:

• Tables list input-output pairs in two columns. Look for patterns (constant differences suggest linear; constant second differences suggest quadratic).
• Graphs plot points $(x, y)$ on a coordinate plane. Analyze shape and key features to identify the function type.
• Equations express the relationship using mathematical symbols. The form of the equation tells you the type (e.g., highest power of $x$ tells you the degree of a polynomial).

Function operations and composition

You can combine functions using arithmetic, just like numbers:

• Addition: $(f + g)(x) = f(x) + g(x)$
• Subtraction: $(f - g)(x) = f(x) - g(x)$
• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$
• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$

Composition chains two functions together. The notation $(f \circ g)(x) = f(g(x))$ means:

1. Evaluate $g(x)$ first
2. Take that result and plug it into $f$

For example, if $f(x) = x^2$ and $g(x) = x + 3$, then $(f \circ g)(x) = f(x+3) = (x+3)^2$.

Watch the domain: the domain of $f \circ g$ is all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$. Also note that composition is not commutative: $f(g(x))$ and $g(f(x))$ are generally different functions.

Symmetry in functions

• Even functions satisfy $f(-x) = f(x)$ for all $x$ in the domain. Their graphs are symmetric about the y-axis. If the point $(a, b)$ is on the graph, so is $(-a, b)$.
  • Examples: $f(x) = x^2$, $f(x) = \cos(x)$
• Odd functions satisfy $f(-x) = -f(x)$ for all $x$ in the domain. Their graphs are symmetric about the origin. If $(a, b)$ is on the graph, so is $(-a, -b)$.
  • Examples: $f(x) = x^3$, $f(x) = \sin(x)$
• Neither even nor odd: most functions fall here. For instance, $f(x) = x^2 + x$ fails both tests since $f(-x) = x^2 - x$, which equals neither $f(x)$ nor $-f(x)$.

To test algebraically, compute $f(-x)$ and compare it to $f(x)$ and $-f(x)$.

Function Transformations and Continuity

Transformations modify a base function's graph in predictable ways:

• Vertical shift: $f(x) + k$ moves the graph up $k$ units (down if $k < 0$)
• Horizontal shift: $f(x - h)$ moves the graph right $h$ units (left if $h < 0$)
• Vertical stretch/compression: $af(x)$ stretches vertically if $|a| > 1$, compresses if $0 < |a| < 1$
• Horizontal stretch/compression: $f(bx)$ compresses horizontally if $|b| > 1$, stretches if $0 < |b| < 1$
• Reflections: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis

A common mistake with horizontal transformations: they work opposite to what you might expect. In $f(x - 3)$, the graph shifts right 3, not left. In $f(2x)$, the graph gets narrower, not wider.

Continuity means a function's graph has no breaks, holes, or jumps. A function $f(x)$ is continuous at a point $a$ if all three conditions hold:

1. $f(a)$ is defined
2. $\lim_{x \to a} f(x)$ exists
3. $\lim_{x \to a} f(x) = f(a)$

A function is continuous on an interval if it's continuous at every point in that interval. Polynomials, for example, are continuous everywhere.

When continuity fails, you get a discontinuity. The three types are:

• Removable (hole): The limit exists, but the function is either undefined or has a different value at that point. You could "fill in" the hole to fix it.
• Jump: The function abruptly changes from one value to another. The left-hand and right-hand limits exist but aren't equal.
• Infinite: The function blows up toward $\pm\infty$ near that point (think vertical asymptotes).

Inverse functions reverse the input-output relationship. For a function $f$, its inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Graphically, the inverse is a reflection of the original over the line $y = x$. A function must be one-to-one (pass the horizontal line test) to have an inverse.