Functions are one of the most fundamental concepts in algebra, and they become even more crucial in Algebra II. At its core, a function is a relationship between inputs and outputs, where each input corresponds to exactly one output.
Note
A function can be thought of as a machine that takes in a value (input) and produces a unique result (output).
In mathematical notation, we often write functions like this:
$f(x) = y$
Where:
Example
For the function $f(x) = x^2$:
A function is one-to-one (or injective) if each element of the range corresponds to at most one element of the domain.
Tip
To test if a function is one-to-one, use the horizontal line test: if a horizontal line intersects the graph of the function at most once, the function is one-to-one.
A function is onto (or surjective) if every element in the range corresponds to at least one element in the domain.
A function that is both one-to-one and onto is called bijective. These functions are particularly important because they have inverses.
Linear functions have a constant rate of change and can be written in the form:
$f(x) = mx + b$
Where $m$ is the slope and $b$ is the y-intercept.
Quadratic functions have a degree of 2 and can be written in the form:
$f(x) = ax^2 + bx + c$
Where $a$, $b$, and $c$ are constants and $a \neq 0$.
Exponential functions have a variable in the exponent and can be written as:
$f(x) = a \cdot b^x$
Where $a$ and $b$ are constants and $b > 0$, $b \neq 1$.
Logarithmic functions are the inverse of exponential functions:
$f(x) = \log_b(x)$
Where $b$ is the base of the logarithm.
Function composition involves applying one function to the result of another:
$(f \circ g)(x) = f(g(x))$
Example
If $f(x) = x^2$ and $g(x) = x + 1$, then: $(f \circ g)(x) = f(g(x)) = f(x + 1) = (x + 1)^2$
The inverse of a function "undoes" what the original function does:
If $f(x) = y$, then $f^{-1}(y) = x$
Note
Not all functions have inverses. Only one-to-one functions have inverses.
Functions can be transformed by:
End behavior describes what happens to $y$ as $x$ approaches positive or negative infinity.
Example
For $f(x) = x^3$: As $x \to \infty$, $f(x) \to \infty$ As $x \to -\infty$, $f(x) \to -\infty$
Zeros (or roots) of a function are the x-values where $f(x) = 0$.
Common Mistake
Don't confuse zeros with x-intercepts. While they're often the same, this is not always true for all functions.
Functions are a powerful tool in mathematics, allowing us to model real-world situations and solve complex problems. As you progress in Algebra II, you'll encounter more advanced function concepts and learn to analyze them in greater depth. Remember, practice is key to mastering these concepts!