1.2 Basic Classes of Functions

Big idea

Calculus is a small set of operations applied to a small catalogue of functions. This section is the catalogue: linear functions, powers, polynomials, rational functions, algebraic functions built from roots, and piecewise definitions that stitch several formulas together. Almost everything you differentiate this term is one of these or a combination of them, so knowing the shape of each family in advance means you can predict what an answer should look like before you compute it.

The second theme is that families are cheap to extend. Once you know the graph of $y = x^2$ or $y = \sqrt{x}$, you know the graph of every shifted, stretched, flipped version of it without plotting a single point. Transformations turn one curve into an infinite supply, and they explain why so many later formulas carry a $(x - a)$ or a leading constant: those are a base function moved into position.

Piecewise functions matter for a specific reason. They are the cheapest way to build a function that behaves differently on different intervals, which makes them the standard source of examples where a limit fails to exist or a derivative fails to exist at a single point. When the next chapter asks you to check the two sides of a point separately, it is usually a piecewise function doing the asking.

End behavior is the long-distance view. For a polynomial, only the leading term matters once $x$ is large, because it outgrows everything beneath it. That single observation classifies the four possible shapes of a polynomial’s tails from two pieces of data, the degree and the sign of the leading coefficient, and it is the first taste of the limits at infinity you will compute formally later.

Decoder

A change applied to the output of a function moves the graph vertically and does exactly what it looks like; a change applied to the input moves the graph horizontally and does the opposite of what it looks like.

Take $y = f(x) + 3$. The change is outside, applied after $f$ has run, so every output rises by $3$ and the graph shifts up by $3$. Take $y = f(x - 3)$. The change is inside, applied to the input before $f$ runs, and to get the old output you now need an input that is $3$ larger, so the graph shifts right by $3$ even though the formula shows a minus sign.

The same reversal governs stretches. Outside, $y = 2f(x)$ doubles every height: a vertical stretch by $2$. Inside, $y = f(2x)$ reaches each old output at half the input, compressing the graph horizontally by a factor of $2$. Reflections follow the pattern too: $-f(x)$ flips across the $x$-axis, $f(-x)$ flips across the $y$-axis.

The classic mistake is misreading a horizontal stretch when a constant has not been factored out. In $y = f(2x - 6)$ the shift is not $6$. Factor first, $f\big(2(x-3)\big)$, and the transformation reads as a horizontal compression by $2$ together with a shift right by $3$.

Definitions and results

Linear functions. A function of the form $f(x) = mx + b$ has a straight-line graph with slope $m$ and $y$-intercept $b$. The slope is the constant rate of change: every increase of $1$ in $x$ changes the output by $m$. Through two points, $m = (y_2 - y_1)/(x_2 - x_1)$.

Polynomials. A polynomial is a finite sum $a_n x^n + \cdots + a_1 x + a_0$ with real coefficients and nonnegative integer exponents. The degree $n$ is the largest exponent with a nonzero coefficient, and $a_n$ is the leading coefficient. Every polynomial has domain all real numbers.

End behavior. For large $|x|$, a polynomial behaves like its leading term $a_n x^n$. If $n$ is even, both tails go the same way: up when $a_n > 0$, down when $a_n < 0$. If $n$ is odd, the tails go opposite ways, up on the right and down on the left when $a_n > 0$, reversed when $a_n < 0$.

Power and root functions. A power function is $f(x) = x^{a}$ for a fixed exponent. Even positive integer powers give $U$-shaped even graphs; odd ones give increasing odd graphs. Writing $\sqrt[n]{x} = x^{1/n}$ puts roots in the same family, with domain $x \ge 0$ when $n$ is even.

Rational and algebraic functions. A rational function is a quotient $p(x)/q(x)$ of polynomials, with domain every $x$ where $q(x) \ne 0$. An algebraic function is anything built from polynomials using the four arithmetic operations and roots. Everything outside that description - sine, $e^x$, $\ln x$ - is called transcendental.

Piecewise definitions. A piecewise function lists a formula for each part of its domain, and the parts must not overlap. To evaluate, first decide which interval the input falls in, then use only that formula.

The transformation rules. Starting from $y = f(x)$, the graph of

$$ y = A\, f\big(B(x - C)\big) + D $$

is stretched vertically by $|A|$ (flipped if $A < 0$), compressed horizontally by $|B|$ (flipped if $B < 0$), then shifted right by $C$ and up by $D$.

Order of operations on a graph. Apply stretches and reflections before shifts. Reversing the order moves the graph to the wrong place, because stretching a shifted graph also scales the shift.

Worked examples

Reading a transformed parabola

Describe the graph of $y = -2(x - 3)^2 + 5$ starting from $y = x^2$.

The inside change is $x - 3$: shift right by $3$. Outside, the factor $-2$ stretches vertically by $2$ and flips across the horizontal direction, and the $+5$ lifts the result by $5$. Applying stretch and reflection first, then the shifts, puts the vertex at $(3, 5)$ opening downward.

Check a point. At $x = 4$ the formula gives $-2(1) + 5 = 3$, one unit right of the vertex and two units down from it, which is exactly the point $(1,1)$ on $y = x^2$ after a doubling, a flip, and the shift.

Evaluating and joining a piecewise function

Let

$$ f(x) = \begin{cases} x^2 & x < 1 \\ 4 - x & x \ge 1 \end{cases} $$

To evaluate, choose the interval first. Since $-2 < 1$, use the top formula: $f(-2) = 4$. Since $1 \ge 1$, use the bottom formula: $f(1) = 3$. Likewise $f(4) = 0$.

Now look at the joint. Approaching $x = 1$ from the left, the values $x^2$ climb toward $1$. The value actually assigned at $x = 1$ is $3$. The two do not agree, so the graph jumps at $x = 1$, and that gap is exactly the situation the next chapter calls a discontinuity. Nothing is wrong with the function; a piecewise rule is under no obligation to meet itself.

End behavior from two numbers

Describe the tails of $p(x) = -3x^5 + 7x^2 - 1$.

Degree $5$ is odd and the leading coefficient $-3$ is negative, so the tails run opposite ways with the right tail going down: $p(x) \to -\infty$ as $x \to \infty$, and $p(x) \to \infty$ as $x \to -\infty$.

Confirm with a large input. At $x = 10$, the leading term is $-3(100000) = -300000$ while the rest contributes $700 - 1 = 699$, a correction of about two tenths of one percent. The leading term is not merely bigger, it is bigger by a growing factor, which is why the lower terms cannot change the direction of a tail.

Domain and shape of a root function

Let $g(x) = 1 + \sqrt{2 - x}$.

Rewrite the inside as $-(x - 2)$. Starting from $y = \sqrt{x}$, the factor $-1$ on the input reflects the graph across the vertical direction, the $-2$ inside shifts it right by $2$, and the $+1$ outside lifts it by $1$.

The square root needs $2 - x \ge 0$, so the domain is $x \le 2$. The root contributes values in $[0, \infty)$, so the range is $[1, \infty)$, with the lowest point $g(2) = 1$. Check one more value: $g(-2) = 1 + \sqrt{4} = 3$, which sits above the corner as the reflected shape predicts.

Practice

Begin with moving a known graph. Identify the base function, then read off each stretch, reflection and shift in the right order.

Next, functions defined in pieces. Decide which interval the input belongs to before touching a formula, and watch what happens at the points where the rule changes.

Last, the long-distance view of a polynomial. Degree and leading sign are the only two inputs you need.

Quiz

Six items on graph transformations, evaluating and analyzing piecewise functions, and polynomial end behavior.