Section 1.1: Four Ways to Represent a Function

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MATH161

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What is This Section About?

Welcome to one of the most fundamental concepts in all of mathematics: functions! You’ve likely used functions before without even realizing it—whenever you follow a recipe, check the temperature, or calculate a tip, you’re working with functions. But what makes functions so powerful in mathematics is that we can represent them in multiple ways, each offering unique insights.

In this section, we explore the four ways to represent a function: verbally (describing it in words), numerically (using tables of values), graphically (plotting points on a coordinate plane), and algebraically (writing formulas). Understanding these different representations is like having four different lenses to examine the same mathematical relationship—each view reveals something important.

Why does this matter? Because real-world problems rarely come pre-packaged in the form you need. You might start with data in a table and need to find a formula. Or you might have an equation and need to visualize its behavior. Mastering these four representations gives you the flexibility to tackle problems from any angle.

The Big Picture: What You’ll Learn

By the end of this section, you’ll be able to:

1. Define what a function is and distinguish it from other mathematical relations
2. Convert between the four representations of functions fluently
3. Determine the domain and range of functions, including those with restrictions from radicals and denominators
4. Apply the Vertical Line Test to identify which curves represent functions
5. Calculate and simplify difference quotients, laying groundwork for derivatives
6. Analyze piecewise defined functions and understand their graphical behavior
7. Recognize even and odd functions through their symmetry properties
8. Identify increasing and decreasing behavior from both graphs and formulas

Think of this section as building your “function vocabulary”—you’re learning multiple languages for describing the same mathematical ideas.

Core Concepts

What is a Function?

At its heart, a function is a rule that assigns to each input exactly one output. This “exactly one” part is crucial—functions are deterministic, meaning the same input always produces the same output.

In plain English: A function is like a machine: you put something in, and it gives you exactly one thing out. Every time you put the same thing in, you get the same thing out.

Key Insight: Not every relationship is a function! For example, “the students in this class” is not a function of “age” because multiple students might be 20 years old. But “age” is a function of “student” because each student has exactly one age.

The Four Representations

1. Verbal (Descriptive)

You describe the function in words. Example: “The area of a circle is pi times the square of its radius.”

When to use it: When explaining concepts or translating real-world situations into mathematics.

2. Numerical (Tables)

You list input-output pairs in a table.

When to use it: When you have data points or want to check specific values quickly.

3. Graphical (Plots)

You plot the function on a coordinate plane, showing the relationship visually.

When to use it: When you want to understand the overall behavior, identify patterns, or communicate visually.

4. Algebraic (Formulas)

You write a formula like $f(x) = 2x - 1$.

When to use it: When you need precision, want to perform calculations, or analyze the function analytically.

Key Takeaway: Each representation has strengths. Tables give specific values, graphs show overall behavior, formulas enable calculations, and verbal descriptions provide context. Learn to move fluidly between them!

Domain and Range: Where Does the Function Live?

The domain is the set of all possible input values (all the $x$-values where the function is defined).

The range is the set of all possible output values (all the $y$-values the function can produce).

Common restrictions to watch for:

1. Square roots: The radicand must be non-negative
   • For $f(x) = \sqrt{x+2}$, we need $x + 2 \geq 0$, so domain is $x \geq -2$
2. Denominators: Cannot divide by zero
   • For $g(x) = \frac{1}{x(x-1)}$, we need $x \neq 0$ and $x \neq 1$
3. Logarithms: Argument must be positive
   • For $h(x) = \ln(x-3)$, we need $x - 3 > 0$, so domain is $x > 3$

Thought Process: Always ask yourself: “What values of $x$ would cause problems in this formula?”

The Vertical Line Test

The test: A curve in the $xy$-plane represents a function if and only if no vertical line intersects it more than once.

Why it works: Remember, a function assigns exactly one output to each input. A vertical line represents all points with the same $x$-coordinate. If that line hits the curve twice, then one $x$-value is producing two different $y$-values—not a function!

Practical tip: Imagine sliding a vertical ruler across the graph. If it ever touches two points simultaneously, you don’t have a function.

The Difference Quotient: Your First Step Toward Calculus

The difference quotient is: \(\frac{f(a+h) - f(a)}{h}\)

This expression measures the average rate of change of $f$ over the interval from $a$ to $a+h$.

In plain English: It’s the slope of the line connecting two points on the function. As $h$ gets smaller, this becomes the instantaneous rate of change—the derivative!

How to compute it:

1. Substitute $a+h$ into $f$ to get $f(a+h)$
2. Subtract $f(a)$ from this
3. Divide the result by $h$
4. Simplify and cancel $h$ (being careful that $h \neq 0$)

Common mistake: Don’t forget to subtract ALL of $f(a)$, including handling negative signs carefully when expanding.

Piecewise Defined Functions

Sometimes functions have different formulas on different parts of their domain. These are called piecewise functions.

Example: \(f(x) = \begin{cases} 1-x, & x \leq -1 \\ x^2, & x > -1 \end{cases}\)

Graphing strategy:

1. Graph each piece separately on its specified interval
2. Pay attention to endpoints—use solid dots for included points ($\leq$ or $\geq$) and open circles for excluded points ($<$ or $>$)
3. Check if the pieces connect smoothly or have jumps (discontinuities)

Even and Odd Functions: Symmetry Matters

Functions can have special symmetry properties:

Even functions: $f(-x) = f(x)$ for all $x$

• Graph symmetry: Symmetric about the $y$-axis (like a mirror)
• Examples: $f(x) = x^2$, $f(x) = \vert x\vert $, $f(x) = \cos x$

Odd functions: $f(-x) = -f(x)$ for all $x$

• Graph symmetry: Symmetric about the origin (180° rotational symmetry)
• Examples: $f(x) = x^3$, $f(x) = \sin x$, $f(x) = x^5 + x$

Quick test: Substitute $-x$ into the formula and simplify. Compare the result to $f(x)$ and $-f(x)$.

Increasing and Decreasing Functions

A function is increasing on an interval if larger inputs produce larger outputs: when $x_1 < x_2$, we have $f(x_1) < f(x_2)$.

A function is decreasing on an interval if larger inputs produce smaller outputs: when $x_1 < x_2$, we have $f(x_1) > f(x_2)$.

Visual check: On a graph, increasing functions go upward as you move left to right, while decreasing functions go downward.

Worked Examples

Example 1: Converting Between Representations

Problem: Consider the function $f(x) = 2x - 1$. (a) Create a table of values for $x = -1, 0, 1, 2$ (b) Describe the function verbally (c) Sketch its graph (d) Find the domain and range

Thought Process: This is a linear function (degree 1 polynomial), so it will have a straight-line graph. Linear functions are defined for all real numbers.

Solution:

(a) Numerical representation:

(b) Verbal representation: “The function doubles the input and then subtracts 1.”

(c) Graphical representation: This is a line with slope 2 and $y$-intercept -1. Plot the points from the table and connect them with a straight line.

(d) Domain and range: Since there are no restrictions (no radicals, denominators, etc.), both domain and range are all real numbers: $(-\infty, \infty)$.

Key Takeaway: For linear functions, all four representations are straightforward. The real power comes when dealing with more complex functions where each representation reveals different information.

Example 2: Finding Domain with Restrictions

Problem: Find the domain of $f(x) = \frac{1}{x^2 - x}$.

Thought Process: This is a rational function (ratio of polynomials). The only restriction is that we cannot divide by zero. So I need to find where the denominator equals zero and exclude those values.

Solution:

Set the denominator equal to zero: \(x^2 - x = 0\) \(x(x-1) = 0\) \(x = 0 \text{ or } x = 1\)

These are the values we must exclude from the domain.

Domain: All real numbers except 0 and 1, written as: \((-\infty, 0) \cup (0, 1) \cup (1, \infty)\)

or in set-builder notation: ${x \in \mathbb{R} : x \neq 0, x \neq 1}$

Key Takeaway: For rational functions, always factor the denominator completely to find all the values that make it zero. These are the “forbidden” values that create vertical asymptotes.

Example 3: Simplifying a Difference Quotient

Problem: For $f(x) = 2x^2 - 5x + 1$, evaluate and simplify: \(\frac{f(a+h) - f(a)}{h}\)

Thought Process: This requires careful algebra. I’ll substitute $a+h$ into the function, expand everything, subtract $f(a)$, and then factor out $h$ from the numerator so I can cancel it with the denominator.

Solution:

Step 1: Find $f(a+h)$ \(f(a+h) = 2(a+h)^2 - 5(a+h) + 1\) \(= 2(a^2 + 2ah + h^2) - 5a - 5h + 1\) \(= 2a^2 + 4ah + 2h^2 - 5a - 5h + 1\)

Step 2: Find $f(a)$ \(f(a) = 2a^2 - 5a + 1\)

Step 3: Compute $f(a+h) - f(a)$ \(f(a+h) - f(a) = (2a^2 + 4ah + 2h^2 - 5a - 5h + 1) - (2a^2 - 5a + 1)\) \(= 4ah + 2h^2 - 5h\)

Step 4: Form the difference quotient and simplify \(\frac{f(a+h) - f(a)}{h} = \frac{4ah + 2h^2 - 5h}{h} = \frac{h(4a + 2h - 5)}{h} = 4a + 2h - 5\)

(Note: We can cancel $h$ because $h \neq 0$ in the difference quotient context)

Final answer: $4a + 2h - 5$

Key Takeaway: Take your time with difference quotients. Write out every step, and watch your signs carefully when subtracting. This tedious algebra now will make derivatives much easier later!

Example 4: Analyzing a Piecewise Function

Problem: For the piecewise function \(f(x) = \begin{cases} 1-x, & x \leq -1 \\ x^2, & x > -1 \end{cases}\) Evaluate $f(-2)$, $f(-1)$, and $f(0)$, and describe how to sketch the graph.

Thought Process: I need to check which piece of the function applies for each input value. The boundary is at $x = -1$.

Solution:

For $f(-2)$: Since $-2 \leq -1$, use the first piece: \(f(-2) = 1 - (-2) = 3\)

For $f(-1)$: Since $-1 \leq -1$ (the boundary is included in the first piece), use the first piece: \(f(-1) = 1 - (-1) = 2\)

For $f(0)$: Since $0 > -1$, use the second piece: \(f(0) = 0^2 = 0\)

Graphing strategy:

1. For $x \leq -1$: Graph the line $y = 1-x$ (slope -1, $y$-intercept 1), but only for $x \leq -1$. Put a solid dot at $(-1, 2)$ since this point is included.

2. For $x > -1$: Graph the parabola $y = x^2$, but only for $x > -1$. Put an open circle at $(-1, 1)$ since this point is NOT included.

3. Notice there’s a “jump” at $x = -1$—the function jumps from the point $(-1, 2)$ down to start the parabola approaching the point $(-1, 1)$.

Key Takeaway: Piecewise functions require you to be a detective—always check which piece applies to your input value. Pay close attention to whether boundary points use $\leq$ (included) or $<$ (excluded).

Practice Problems

Try these problems to solidify your understanding:

1. Domain Practice: Find the domain of $f(x) = \sqrt{x+2}$.

2. Vertical Line Test: Does the curve defined by $x = y^2 - 2$ represent a function? Explain using the Vertical Line Test.

3. Difference Quotient: For $f(x) = 4 + 3x - x^2$, evaluate and simplify $\frac{f(x+h) - f(x)}{h}$.

4. Even or Odd? Determine whether $h(x) = 2x - x^2$ is even, odd, or neither.

5. Piecewise Evaluation: For $g(x) = \begin{cases} x+1, & x < 0 \ x^2+1, & x \geq 0 \end{cases}$, find $g(-1)$, $g(0)$, and $g(2)$.

6. Multiple Representations: The function $f$ satisfies $f(0) = 3$, $f(1) = 5$, and $f(2) = 7$. Find an algebraic formula for $f(x)$.

Hints:

• For problem 1: What values make the radicand non-negative?
• For problem 2: Try drawing vertical lines—does any line hit the curve twice?
• For problem 3: Expand carefully and factor out $h$ before canceling
• For problem 4: Compute $h(-x)$ and compare to $h(x)$ and $-h(x)$
• For problem 5: Check the condition for each value before choosing which formula to use
• For problem 6: The pattern suggests a linear function—find the slope and intercept

Key Reminders

As you work with functions, keep these strategies in mind:

✓ Think “input-output” – Functions are machines that transform inputs into outputs consistently

✓ Check domain restrictions – Look for radicals, denominators, and logarithms that impose constraints

✓ Use multiple representations – If you’re stuck with one view, try converting to another representation

✓ Master the difference quotient – This algebraic skill is essential for calculus success

✓ Pay attention to endpoints – Piecewise functions require careful attention to $\leq$ vs. $<$ symbols

✓ Visualize symmetry – Even and odd functions have predictable graph behavior

Why This Matters

Functions are the language of mathematics and science. Every time you:

• Use GPS navigation (position as a function of time)
• Check stock prices (price as a function of time)
• Adjust a recipe (ingredients as a function of servings)
• Calculate compound interest (balance as a function of time)

…you’re working with functions! Understanding how to represent and manipulate functions gives you tools to model virtually any real-world relationship. The four representations you’ve learned are like having multiple tools in your mathematical toolbox—each one perfect for different situations.