1.2 Basic Classes of Functions

Every function you'll encounter in Calculus I belongs to one of a few basic classes. Recognizing which class a function belongs to tells you a lot about its shape, behavior, and domain before you ever take a derivative. This section covers linear functions, polynomials, and several other important function types, along with the operations and properties that apply across all of them.

Linear Functions

Slope of linear functions

The slope of a line captures its rate of change: how much $y$ changes for each unit increase in $x$. For any two points on the line, slope is calculated as:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

This value stays constant everywhere on a linear function, which is exactly what makes it linear.

• Positive slope: the line rises from left to right (increasing function)
• Negative slope: the line falls from left to right (decreasing function)
• Zero slope: the line is horizontal (constant function)
• Undefined slope: the line is vertical (not a function)

The steepness of the line corresponds to the absolute value of the slope. A slope of $-5$ is steeper than a slope of $2$, because $|-5| > |2|$.

Polynomial Functions

Polynomial function characteristics

A polynomial function is a sum of terms, each consisting of a coefficient times a variable raised to a non-negative integer exponent. The general form is:

$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

The degree is the highest exponent that appears (with a nonzero coefficient). Degree determines the function's overall shape:

• Degree 1 (linear): $ax + b$
• Degree 2 (quadratic): $ax^2 + bx + c$
• Degree 3 (cubic): $ax^3 + bx^2 + cx + d$

End behavior depends on the degree and the sign of the leading coefficient ($a_n$):

• Odd-degree polynomials always have opposite end behavior. If the leading coefficient is positive, the graph falls to the left and rises to the right. They're guaranteed at least one real root.
• Even-degree polynomials have the same end behavior on both sides. A positive leading coefficient means both ends rise toward $+\infty$; a negative one means both ends fall toward $-\infty$. They may have no real roots at all.

Quadratic equations and roots

A quadratic equation has the form $ax^2 + bx + c = 0$ where $a \neq 0$. Its roots (the $x$-values where the parabola crosses the $x$-axis) are found with the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The expression under the square root, $b^2 - 4ac$, is called the discriminant, and it tells you what kind of roots to expect:

• Positive discriminant: two distinct real roots (parabola crosses the $x$-axis twice)
• Zero discriminant: one repeated real root (parabola just touches the $x$-axis)
• Negative discriminant: no real roots, two complex conjugate roots (parabola doesn't reach the $x$-axis)

For example, $2x^2 + 3x - 5 = 0$ has discriminant $9 - 4(2)(-5) = 49$, which is positive, so there are two distinct real roots.

Other Functions

Types of algebraic functions

Rational functions are ratios of two polynomials, like $f(x) = \frac{x+1}{x^2 - 4}$. They can have:

• Vertical asymptotes where the denominator equals zero (and the numerator doesn't), such as $x = 2$ and $x = -2$ in the example above
• Horizontal asymptotes that describe the function's behavior as $x \to \pm\infty$

Power functions have the form $f(x) = x^a$ where $a$ is any real constant. Their symmetry depends on $a$:

• Even integer $a$: symmetric about the $y$-axis (e.g., $x^2, x^4$)
• Odd integer $a$: symmetric about the origin (e.g., $x^3, x^5$)
• Negative $a$: the function decreases on its domain and produces hyperbolic shapes (e.g., $x^{-1} = \frac{1}{x}$)

Root functions have the form $f(x) = \sqrt[n]{x} = x^{1/n}$. For even $n$, the domain is $[0, \infty)$. For odd $n$ (like cube roots), the domain is all real numbers. Their graphs are increasing and concave down on $(0, \infty)$.

Algebraic vs. transcendental functions

Algebraic functions are built from a finite number of algebraic operations: addition, subtraction, multiplication, division, and taking roots. Polynomials, rational functions, power functions, and root functions all fall in this category.

Transcendental functions are everything else. The main types you'll use in Calculus I are:

• Exponential (e.g., $2^x$, $e^x$)
• Logarithmic (e.g., $\ln x$, $\log_{10} x$)
• Trigonometric (e.g., $\sin x$, $\cos x$)

These often have distinctive features like asymptotes (logarithms), periodicity (trig functions), or unbounded growth (exponentials) that set them apart from algebraic functions.

Piecewise function graphing

A piecewise function uses different formulas on different intervals of its domain. To graph one:

1. Identify each interval and its corresponding expression
2. Graph each piece only on its assigned interval
3. Mark endpoints with a closed circle (●) if that endpoint is included, or an open circle (○) if it's excluded

Pay close attention to where the pieces meet. If the function values don't match at a boundary, there's a jump discontinuity at that point.

Function graph transformations

Starting from the graph of $y = f(x)$, you can shift, stretch, compress, or reflect it:

Vertical shifts:

• $f(x) + k$: shifts up by $k$ units
• $f(x) - k$: shifts down by $k$ units

Horizontal shifts:

• $f(x - h)$: shifts right by $h$ units
• $f(x + h)$: shifts left by $h$ units

A common mistake: horizontal shifts go in the opposite direction from what the sign suggests. $f(x - 3)$ shifts right, not left.

Vertical stretches/compressions:

• $af(x)$ with $|a| > 1$: stretches vertically (taller)
• $af(x)$ with $0 < |a| < 1$: compresses vertically (shorter)

Horizontal stretches/compressions:

• $f(bx)$ with $|b| > 1$: compresses horizontally (narrower)
• $f(bx)$ with $0 < |b| < 1$: stretches horizontally (wider)

Reflections:

• $-f(x)$: reflects over the $x$-axis
• $f(-x)$: reflects over the $y$-axis

Function Properties and Operations

Domain and range

The domain is the set of all valid input values ($x$-values) for a function. The range is the set of all output values ($y$-values) the function actually produces.

When finding domains, watch for two main restrictions: you can't divide by zero, and you can't take an even root of a negative number. For example, $f(x) = \frac{1}{\sqrt{x - 3}}$ requires $x - 3 > 0$, so the domain is $(3, \infty)$.

Function composition and inverse

Composition feeds the output of one function into another. The notation $(f \circ g)(x)$ means $f(g(x))$: first apply $g$, then apply $f$ to the result. Order matters here, since $f(g(x))$ and $g(f(x))$ are generally different.

An inverse function $f^{-1}(x)$ reverses what $f$ does. If $f(2) = 5$, then $f^{-1}(5) = 2$. Not every function has an inverse. A function must be one-to-one (pass the horizontal line test) to have an inverse. Graphically, the inverse is a reflection of the original function over the line $y = x$.

Continuity and limits

A function is continuous at a point if you can draw through that point without lifting your pencil. More precisely, $f$ is continuous at $x = a$ when three things hold: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

Discontinuities come in several types: holes (removable), jumps, and vertical asymptotes (infinite). Polynomials are continuous everywhere, but rational and piecewise functions often have points of discontinuity.

A limit describes what value $f(x)$ approaches as $x$ gets close to some number, even if the function isn't defined there. This concept is the bridge between the algebra-heavy material in this unit and the core ideas of calculus that come next.