Welcome to Function Notation: Your Mathematical Recipe Book!

Hey there! Ready to tackle one of the most fundamental ideas in higher mathematics? Function notation might look a little scary at first—it uses letters you might not have seen together before—but don't worry! It’s actually just a very efficient way to write down a mathematical rule.

Think of this chapter as learning how to read a concise recipe. Instead of writing "Take a number, multiply it by two, and add one," we use notation to write that rule clearly and quickly.

Why is this important? Functions are the backbone of sequences, graphs, and calculus. Mastering this notation makes reading complex problems much easier and sets you up for success in future topics!

1. What is a Function? (The Function Machine Analogy)

Before we dive into the notation, let’s quickly remind ourselves what a function is.

A Function is a rule that takes an input, processes it, and produces exactly one output.

Imagine a Function Machine:

  • You put a number (the Input) into the machine.
  • The machine follows a specific rule (e.g., 'square the number').
  • It spits out one resulting number (the Output).

Example: If the rule is "add 5 to the input," and you put in 3, the output is 8.

Key Terms in Functions

Input Variable: Usually represented by \(x\). This is the value you start with.
Output Variable: Usually represented by \(y\). This is the result of the function.
The Rule: The expression that tells you what to do with the input (e.g., \(2x + 1\)).

Key Takeaway: A function is a reliable rule where every input gives only one output.

2. Introducing Function Notation: \(f(x)\)

Instead of writing "let \(y\) be equal to the rule applied to \(x\)", we use function notation.

The standard notation is \(f(x)\).

What does \(f(x)\) mean?

The notation \(f(x)\) is read as "f of x". It is simply a fancy way of saying the output \(y\) that results when the function \(f\) operates on the input \(x\).

If you have the equation \(y = 2x + 1\), you can rewrite this using function notation as:

\[ f(x) = 2x + 1 \]

Important Tip! A Common Mistake to Avoid:

The notation \(f(x)\) DOES NOT mean \(f\) multiplied by \(x\). It is a single symbol representing the output value.

Why do we use different letters?

Sometimes you might see \(g(x)\) or \(h(t)\). The letter outside the bracket (like \(f\), \(g\), or \(h\)) just names the function, especially when you are dealing with more than one rule at a time.

  • \(f(x)\): Function f uses input \(x\).
  • \(g(x)\): Function g uses input \(x\).
  • \(h(t)\): Function h uses input \(t\).

Did you know? Functions are critical when plotting graphs! When you plot a graph of a function, you are plotting the points \((x, f(x))\).

3. Evaluating Functions (Finding the Output)

Evaluating a function means finding the output when you are given a specific input value.

If you see \(f(4)\), this means: "Use the input value \(x = 4\) in the rule for function \(f\)."

Step-by-Step Process for Evaluation

Let's use the function \(f(x) = 3x - 5\).

Problem: Find \(f(4)\).

  1. Step 1: Identify the Input. The input is \(x = 4\).
  2. Step 2: Replace \(x\) with the Input Value. Replace every \(x\) in the rule with \(4\).

    3. \[ f(4) = 3(4) - 5 \]

  3. Step 3: Calculate the Result.

    4. \[ f(4) = 12 - 5 \]

    \[ f(4) = 7 \]

The output is 7. We can write this as \((4, 7)\).

Example with Negative Numbers

Let \(g(x) = x^2 + 2x\). Find \(g(-3)\).

Crucial Tip: When substituting negative numbers, always use brackets to ensure you apply powers correctly!

\[ g(-3) = (-3)^2 + 2(-3) \]

\[ g(-3) = 9 + (-6) \]

\[ g(-3) = 9 - 6 \]

\[ g(-3) = 3 \]

Quick Review Box:
To evaluate \(f(a)\), simply replace \(x\) with \(a\) in the function’s rule and calculate the result.

4. Finding the Input (Solving for x)

Sometimes, you are given the output and need to find the original input. This requires setting the function rule equal to the given output and solving the resulting equation.

Step-by-Step Process for Finding Input

Let \(h(x) = 5x + 10\).

Problem: Find the value of \(x\) for which \(h(x) = 35\).

  1. Step 1: Set the Rule equal to the Output. Since \(h(x)\) is the output, we replace \(h(x)\) with 35.

    2. \[ 5x + 10 = 35 \]

  2. Step 2: Solve the Equation for \(x\).

    3. Subtract 10 from both sides:

    \[ 5x = 35 - 10 \]

    \[ 5x = 25 \]

    Divide by 5:

    \[ x = 5 \]

The input that gives an output of 35 is \(x = 5\).

Dealing with Quadratic Functions (Finding Two Inputs)

If the function involves \(x^2\), you might find two possible input values.

Let \(f(x) = x^2 - 1\).

Problem: Find the value(s) of \(x\) for which \(f(x) = 8\).

  1. Set up the equation:

    2. \[ x^2 - 1 = 8 \]

  2. Solve for \(x\):

    3. Add 1 to both sides:

    \[ x^2 = 9 \]

    Take the square root (remembering both positive and negative solutions):

    \[ x = \pm \sqrt{9} \]

    \[ x = 3 \text{ or } x = -3 \]

Both \(3\) and \(-3\) give the output 8.

Key Takeaway: Finding the input involves solving a linear or quadratic equation based on the function rule.

5. Substituting Expressions into Functions

Function notation is flexible! The input doesn't always have to be a single number like 4. It can be a variable, an algebraic expression, or even the result of another function.

Example: Substituting an Expression

Let \(f(x) = 2x + 7\). Find an expression for \(f(a+1)\).

We replace every \(x\) in the rule with the entire expression \((a+1)\).

\[ f(a+1) = 2(a+1) + 7 \]

Now, expand and simplify:

\[ f(a+1) = 2a + 2 + 7 \]

\[ f(a+1) = 2a + 9 \]

Example: Substituting another Variable

Let \(g(x) = x^2 - 3\). Find an expression for \(g(y)\).

This is straightforward substitution:

\[ g(y) = y^2 - 3 \]

Don't worry if this seems tricky at first! Remember that whatever is inside the brackets \((...)\) is the new input, and you must substitute the whole thing into the function's rule wherever \(x\) used to be.

6. Review and Practice

You have successfully mastered the foundation of function notation! Remember that \(f(x)\) is just a label for the output \(y\).

Quick Concept Review
  • Notation: \(f(x)\) means "the output of function \(f\) when the input is \(x\)."
  • Evaluation: To find \(f(a)\), substitute \(a\) for \(x\) in the equation.
  • Solving: To find \(x\) when \(f(x) = k\), set the function rule equal to \(k\) and solve for \(x\).

Encouragement: Keep practicing the substitution steps, especially with negative numbers and algebraic expressions. You'll soon find function notation a helpful tool rather than a hurdle!