9.1 Function Analysis

Understanding Function Behavior

In earlier chapters you met the basic idea of a function and worked with many specific types (linear, quadratic, exponential, etc.). Function analysis is about looking at a function as a whole and describing how it behaves on its domain.

In this chapter we focus on the following kinds of questions:

• For which inputs is the function defined? (domain)
• What outputs can it produce? (range)
• Where is it positive or negative?
• Where is it increasing or decreasing?
• How does it behave for very large positive or negative inputs?
• Where does it have peaks, valleys, and changes in behavior?

The goal is to turn a formula or a graph into a clear description of the function’s overall behavior.

Throughout, you should assume you already understand what a function is, function notation, and common basic functions from earlier chapters.

Domain and Range as Tools for Analysis

The domain and range of a function are not only basic definitions; they are also fundamental tools for describing how a function behaves.

We will not re‑define domain and range in full detail here, but we will see how they interact with other aspects of analysis.

Using the domain to understand behavior

Sometimes the most important features of a function come from where it is not defined.

Typical sources of domain restrictions include:

• Denominators: expressions like $\dfrac{1}{x-3}$ are undefined when the denominator is $0$.
• Even roots: expressions like $\sqrt{g(x)}$ (over the real numbers) are only defined when $g(x) \ge 0$.
• Logarithms (covered in more detail in Algebra II): $\log(g(x))$ is defined only when $g(x) > 0$.

When analyzing a function, you should always:

1. Identify its domain.
2. Mark any “problem points” (values excluded from the domain).
3. Note how the function behaves near those problem points.

For example, consider
$$
f(x) = \frac{2x+1}{x-4}.
$$

• The denominator $x-4$ must not be $0$, so $x \ne 4$.
• Thus the domain is “all real numbers except $4$.”
• Near $x=4$, the function grows very large in the positive or negative direction; understanding this is part of function analysis.

Relating domain, range, and a graph

On a graph:

• The domain corresponds to the set of $x$‑values covered along the horizontal axis.
• The range corresponds to the set of $y$‑values the curve reaches along the vertical axis.

Function analysis often involves turning between:

• An algebraic description of domain and range (like “$x \ne 4$” or “$y > -3$”),
• And a graphical description (where gaps, holes, and asymptotes appear visually).

When you analyze a function’s behavior, describing its domain and range precisely is usually one of the first and last steps: you refine your understanding of range as you learn more about the function’s graph and behavior.

Key Features of Graphs

A large part of function analysis is describing the important geometric features of a function’s graph in the coordinate plane.

Here are some of the features commonly analyzed.

Intercepts

• $x$‑intercepts: Points where the graph crosses the $x$‑axis, so $f(x) = 0$.
• $y$‑intercept: Where the graph crosses the $y$‑axis, so $x=0$, and the point is $(0, f(0))$ (when $0$ is in the domain).

Finding and stating intercepts helps anchor the graph and is often the first algebraic step in describing what the graph looks like.

Symmetry

Recognizing symmetry simplifies analysis and graphing.

Common types of symmetry:

• Even symmetry: A function is even if $f(-x) = f(x)$ for all $x$ in the domain.
• Graph is symmetric about the $y$‑axis.
• Example: $f(x) = x^2$.
• Odd symmetry: A function is odd if $f(-x) = -f(x)$ for all $x$ in the domain.
• Graph is symmetric about the origin (a $180^\circ$ rotation gives the same graph).
• Example: $f(x) = x^3$.

Checking for even/odd symmetry is part of function analysis because:

• It can immediately tell you about the function’s behavior on half of the domain once you know it on the other half.
• It often links to algebraic properties, such as whether only even or only odd powers of $x$ appear in a polynomial.

Some functions have other kinds of symmetry (for example, periodic functions repeat every certain interval); these are treated in more detail in trigonometry and function‑specific chapters, but recognizing symmetry is always part of analysis.

Asymptotic behavior (informal view)

You will see a more formal treatment of limits in later chapters. Here we only describe the qualitative idea.

An asymptote is a line that the graph of a function gets closer and closer to, without actually meeting it (at least in some direction).

Common types:

• Vertical asymptotes, like $x = a$, where the function’s values grow without bound as $x$ approaches $a$.
• Horizontal asymptotes, like $y = b$, describing the long‑term value the function approaches as $x$ becomes very large positive or negative.
• Oblique (slant) asymptotes, like $y = mx + c$, where the graph approaches a straight line with non‑zero slope.

In this precalculus view, when analyzing a function you should:

• Identify where the function is undefined and check whether a vertical asymptote is likely.
• Examine the behavior as $x \to \infty$ and $x \to -\infty$ to see if a horizontal or slant asymptote appears.

The precise definitions and tools (limits) will come later; here, it is enough to recognize and describe the patterns.

Discontinuities (informal view)

A function is continuous (roughly) if you can draw its graph without lifting your pencil. When analyzing a function, you often need to know where it fails to be continuous and what kind of failure is happening.

Qualitatively, you might see:

• Holes: a single missing point in an otherwise smooth curve (a “removable” issue).
• Jumps: the graph jumps from one value to another with no connection.
• Asymptotic breaks: where the function shoots off toward infinity near a vertical asymptote.

Function analysis involves:

• Spotting these on a graph.
• Relating them to algebraic issues such as zero denominators or piecewise definitions.

The detailed theory of continuity and limits is reserved for calculus; here, you practice recognizing and describing discontinuities.

Increasing, Decreasing, and Constant Behavior

A central part of function analysis is understanding where a function is going up or down as you move along the $x$‑axis.

Basic definitions (graphical viewpoint)

Let $f$ be a function and consider its graph.

• $f$ is increasing on an interval if, as $x$ moves to the right, the graph moves upward. More precisely: for any two points $x_1 < x_2$ in the interval,
  $$
  f(x_1) < f(x_2).
  $$
• $f$ is decreasing on an interval if, as $x$ moves to the right, the graph moves downward. More precisely: for $x_1 < x_2$ in the interval,
  $$
  f(x_1) > f(x_2).
  $$
• $f$ is constant on an interval if the graph is a horizontal line there. More precisely: for any $x_1, x_2$ in the interval,
  $$
  f(x_1) = f(x_2).
  $$

These definitions relate closely to the idea of monotonicity, which is studied more specifically in the next subchapter. Here, we emphasize how to recognize and describe these patterns.

Describing intervals of increase and decrease

On a graph:

• Identify stretches where the curve slopes upward as you move left to right; these are intervals where the function is increasing.
• Identify stretches where the curve slopes downward; these are intervals where the function is decreasing.
• Identify flat stretches; these are intervals where the function is constant.

You then translate these observations into interval notation, such as:

• “$f$ is increasing on $(1, 3)$ and $(4, \infty)$, decreasing on $(-\infty, 1)$, and constant on $[3,4]$.”

Function analysis practice often involves:

• Starting from a formula, sketching or using known shapes (like for polynomials, exponentials, etc.), then stating where the function goes up or down.
• Or starting from a graph and simply reading off the intervals of increase/decrease.

Later, in calculus, derivatives will give a precise algebraic way to find these intervals. Here, the focus is on understanding and stating them correctly.

Local and Global Extrema

A function’s extrema (plural of “extreme values”) are its highest and lowest points. Analyzing these is crucial in many applications, such as optimization.

Local (relative) maxima and minima

A local maximum is a point where the function is higher than nearby points, even if it’s not the highest value overall. A local minimum is where the function is lower than nearby points.

Graphically:

• At a local maximum, the graph has a peak.
• At a local minimum, the graph has a valley.

These can occur at “turning points” in the graph, where the function changes from increasing to decreasing (for a local maximum) or from decreasing to increasing (for a local minimum).

For function analysis at this level:

• You learn to identify local maxima and minima from a graph.
• You describe them using coordinates, like “$f$ has a local maximum at $(2, 5)$.”

The formal conditions using derivatives are handled in calculus; here, the emphasis is on geometry and intuition.

Global (absolute) maximum and minimum

A global maximum (or absolute maximum) is the highest value the function ever attains on its entire domain. A global minimum is the lowest.

Not every function has a global maximum or minimum. For instance:

• A function like $f(x) = x^2$ has a global minimum at $(0,0)$ but no global maximum, because it grows without bound as $|x|$ grows.
• A function like $f(x) = x^3$ has neither a global maximum nor a global minimum.

In function analysis you should:

• Distinguish between local and global extrema.
• Recognize when the graph suggests that no global extremes exist because the function keeps going up or down indefinitely.

In the special context of closed bounded intervals (like $[a,b]$), continuous functions always have global maxima and minima; this idea becomes important in calculus, but you can already see it visually when analyzing graphs.

End Behavior

End behavior describes what a function does as $x$ becomes very large positive or very large negative.

Informally, we ask questions like:

• As $x \to \infty$, does $f(x)$:
• Approach a finite value?
• Grow without bound?
• Approach $0$?
• Oscillate without settling?
• As $x \to -\infty$, does $f(x)$ behave in the same way or differently?

Describing end behavior is particularly important for:

• Polynomial functions (analyzed more in Algebra II and Precalculus),
• Rational functions (ratios of polynomials),
• Exponential and logarithmic functions,
• Trigonometric functions (which often oscillate).

In this chapter, the key skill is:

• Translating algebraic knowledge of a function into a verbal or graphical description such as:
• “As $x \to \infty$, $f(x)$ grows without bound.”
• “As $x \to -\infty$, $f(x)$ approaches $0$ from above.”
• “As $x \to \pm\infty$, $f(x)$ approaches the horizontal asymptote $y = 3$.”

You will later learn to express these ideas precisely with limits; for now, you should become comfortable seeing and describing them.

Piecewise‑Defined Functions

Many real‑world relationships and many test examples are piecewise‑defined functions: different formulas apply on different parts of the domain.

A typical piecewise function might be defined as something like
$$
f(x) =
\begin{cases}
\text{one expression}, & x < a, \\
\text{another expression}, & x \ge a.
\end{cases}
$$

In function analysis, piecewise functions are important because they show how behavior can change abruptly at certain points.

When analyzing such a function, you should:

1. Consider each piece separately:
• Determine its domain and range (on that piece),
• Describe where it increases or decreases and any local maxima/minima within that piece.
2. Pay special attention to the “junction points” where the pieces meet:
• Is the function defined there?
• Does the left‑hand value match the right‑hand value (graph connects) or is there a jump or hole?
• Is there a sharp corner or cusp, even if the function is continuous?

Function analysis of piecewise functions gives practice in carefully combining information from different regions and is good preparation for later topics like continuity and differentiability.

Qualitative Graph Sketching

Putting all the ideas together, function analysis often culminates in sketching a “rough” graph from algebraic information, or conversely, extracting algebraic and verbal descriptions from a given graph.

A qualitative sketch typically shows:

• The domain (including any excluded points),
• Intercepts,
• Obvious symmetry,
• Rough intervals where the function increases and decreases,
• Local maxima and minima,
• Asymptotic behavior, and
• Approximate end behavior.

The sketch does not need to be perfectly accurate or to scale, but it should reflect the correct overall shape and key features.

When you are given a graph, basic function analysis involves:

• Identifying and stating all the above features clearly.
• Translating between visual features (like peaks, valleys, breaks, and long‑term trends) and formal descriptions (like “local maximum at $x=2$,” “vertical asymptote at $x=4$,” “domain is $\mathbb{R} \setminus \{4\}$,” and so on).

Later, calculus will give you powerful algebraic tools to compute many of these features exactly. Precalculus function analysis prepares you by building the geometric and conceptual understanding you will need.