Introduction to Functions

Welcome to the world of functions! In Mathematics: Applications and Interpretation, functions are the bread and butter of mathematical modelling. Think of a function as a sophisticated "input-output machine." You feed it a number (the input), it performs a specific rule, and it spits out a result (the output). Whether you are calculating the cost of production, the trajectory of a ball, or the growth of a population, you are using functions.

What is a Function?

A function is a special relationship where every input has exactly one output. If you put the same number into the machine twice, you must get the same result both times.

Function Notation

We use specific shorthand to talk about functions. While \(f(x)\) is the most common, IB applications often use letters that make sense in context:

  • \(f(x)\): The value of function \(f\) at input \(x\).
  • \(v(t)\): Velocity (\(v\)) as a function of time (\(t\)).
  • \(C(n)\): Total cost (\(C\)) as a function of the number of items (\(n\)).

Did you know? The graph of a function will always pass the "Vertical Line Test." If you draw a vertical line anywhere on the graph and it hits the curve more than once, it is not a function!

Domain and Range

Understanding a function's "boundaries" is crucial for modelling real-world situations.

1. Domain

The domain is the set of all possible input values (usually the \(x\)-values) for which the function is defined.
Analogy: If a function is a recipe, the domain is the list of ingredients you are allowed to use. You can't make an omelet with a brick!

In real-world models, the domain is often restricted by logic. For example, if \(t\) represents time, the domain is usually \(t \geq 0\).

2. Range

The range is the set of all possible output values (the \(y\)-values) that the function actually produces.

How to identify them:
  • From a graph: Look at how far the graph goes left-to-right for the domain, and how far it goes down-to-up for the range.
  • From an equation: Look for "danger zones" like denominators (which can't be zero) or square roots of negative numbers.

Quick Review:
Domain: How wide is the graph? (\(x\)-axis)
Range: How tall is the graph? (\(y\)-axis)

Inverse Functions

An inverse function essentially "undoes" the original function. If function \(f\) takes you from \(A\) to \(B\), the inverse function \(f^{-1}\) takes you back from \(B\) to \(A\).

Notation and Visualizing

The notation for an inverse is \(f^{-1}(x)\).
Important: \(f^{-1}(x)\) is not the same as \(\frac{1}{f(x)}\). In functions, the \(-1\) is a label, not an exponent!

Graphical Relationship

The graph of an inverse function is a reflection of the original function in the line \(y = x\). This is because the \(x\) and \(y\) coordinates swap places. If the point \((2, 5)\) is on the original graph, then \((5, 2)\) will be on the inverse graph.

Finding the Inverse Algebraically (HL Only)

To find the expression for \(f^{-1}(x)\):

  1. Replace \(f(x)\) with \(y\).
  2. Swap the \(x\) and the \(y\).
  3. Rearrange the equation to solve for \(y\).
  4. Replace the new \(y\) with \(f^{-1}(x)\).

Example: If \(f(x) = 2x + 3\), we write \(y = 2x + 3\). Swap them: \(x = 2y + 3\). Solve for \(y\): \(x - 3 = 2y \implies y = \frac{x - 3}{2}\). So, \(f^{-1}(x) = \frac{x-3}{2}\).

Domain Restrictions (HL Only)

Sometimes, a function doesn't have an inverse because it isn't "one-to-one" (like a parabola \(y = x^2\)). In these cases, we must restrict the domain of the original function (e.g., only looking at \(x \geq 0\)) so that it passes the horizontal line test and an inverse can exist.

Composite Functions (HL Only)

A composite function is created when you apply one function to the result of another. You are "nesting" the functions.

Notation: \((f \circ g)(x)\) or \(f(g(x))\).

To calculate \(f(g(x))\), you always work from the inside out:

  1. Calculate the output of the inner function \(g(x)\).
  2. Use that output as the input for the outer function \(f\).

Context Example: If \(g(d)\) calculates the tax on a dollar amount \(d\), and \(f(t)\) adds a processing fee to a tax amount \(t\), then \(f(g(d))\) calculates the final fee after tax is applied to the original dollars.

Key Takeaways and Common Mistakes

Key Takeaways:

  • Functions assign one output to every input.
  • Domain = \(x\)-values; Range = \(y\)-values.
  • The inverse \(f^{-1}(x)\) reflects the graph over the line \(y = x\).
  • Always check if a model makes sense in context (e.g., negative time or negative distance usually don't exist).

Common Mistakes to Avoid:

  • Don't confuse \(f^{-1}(x)\) with a reciprocal. It's a "backwards" machine, not a fraction!
  • When finding the range from a calculator, make sure your window is zoomed out enough to see the maximum and minimum points.
  • In composite functions \(f(g(x))\), students often do them in the wrong order. Remember: Inner first, then Outer!

Don't worry if domain and range seem tricky at first. The more you look at graphs on your GDC (Graphic Display Calculator), the more intuitive these boundaries will become!