Functions are fundamental objects in mathematics that describe relationships between variables. They form a cornerstone of mathematical analysis and are extensively used in modeling real-world phenomena.
A function $f$ is a rule that assigns to each element $x$ in a set $A$ exactly one element $y$ in a set $B$. We write this as $f: A \to B$ or $y = f(x)$.
Example
The function $f(x) = x^2$ assigns to each real number its square. Here, the domain and codomain are both the set of real numbers, but the range is only the non-negative real numbers.
Note
The notation $f(x)$ is read as "f of x" and represents the output value of the function for a given input $x$.
The graph of a function is the set of all points $(x, y)$ in the coordinate plane such that $y = f(x)$. It provides a visual representation of the function's behavior.
Tip
When sketching graphs, pay attention to key features such as intercepts, maxima/minima, and symmetry. These can often be determined from the function's equation.
For a function $f: A \to B$, its inverse function $f^{-1}: B \to A$ (if it exists) "undoes" what $f$ does. Mathematically, $f^{-1}(f(x)) = x$ for all $x$ in $A$, and $f(f^{-1}(y)) = y$ for all $y$ in $B$.
Example
If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x-3}{2}$.
Common Mistake
Not all functions have inverses. For a function to have an inverse, it must be both injective (one-to-one) and surjective (onto).
To find the inverse of a function algebraically:
Given two functions $f$ and $g$, their composition $(f \circ g)(x)$ is defined as $f(g(x))$. This means we apply $g$ first, then apply $f$ to the result.
Example
If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x+1)^2$.
A quadratic function has the general form $f(x) = ax^2 + bx + c$, where $a \neq 0$. Its graph is a parabola.
Key features:
Example
For $f(x) = x^2 - 4x + 3$:
A rational function is a ratio of polynomials: $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$.
Key features to consider:
Example
For $f(x) = \frac{x^2-1}{x-1}$:
Exponential function: $f(x) = a^x$, where $a > 0$ and $a \neq 1$ Logarithmic function: $f(x) = \log_a(x)$, where $a > 0$ and $a \neq 1$
These functions are inverses of each other.
Note
The natural exponential function $e^x$ and natural logarithm $\ln(x)$ are particularly important in calculus and many applications.
Given a function $y = f(x)$, we can transform its graph in several ways:
Example
Starting with $f(x) = x^2$, the function $g(x) = 2(x-1)^2 + 3$ represents:
Equations can often be solved both graphically and analytically. The graphical method involves finding points of intersection between curves, while the analytical method uses algebraic techniques.
Example
To solve $x^2 = 2x + 3$:
Graphically: Plot $y = x^2$ and $y = 2x + 3$ and find their intersection points.
Analytically: Rearrange to $x^2 - 2x - 3 = 0$, then use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=-2$, $c=-3$, giving solutions $x = 3$ or $x = -1$.
Tip
Graphing calculators or software can be extremely helpful for visualizing functions and solving equations graphically.
A polynomial function of degree $n$ has the form: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $a_n \neq 0$ and $n$ is a non-negative integer.
Key concepts:
Example
For $f(x) = x^3 - x^2 - 4x + 4$:
Graphically:
Example
For non-injective functions, we can often find an inverse by restricting the domain.
Example
$f(x) = x^2$ is not injective on $\mathbb{R}$, but we can define:
Solving inequalities often involves similar techniques to solving equations, but with careful attention to the direction of the inequality when multiplying or dividing by negative numbers.
Example
To solve $\frac{x-1}{x+2} > 3$:
Also need to consider $x \neq -2$ due to the original denominator.
The modulus (or absolute value) function is defined as:
$$|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x
< 0 \end{cases}$$
Graphically, this "reflects" the negative part of a function above the x-axis.
Example
The graph of $y = |x-2|$ is a V-shape with its vertex at (2,0).
Note
Equations involving modulus often require considering multiple cases or squaring both sides.
This comprehensive overview covers the key aspects of functions as studied in the IB Mathematics AA curriculum, providing a solid foundation for further study in calculus and mathematical analysis.
