Mathematics Study Guide for the ACT

Functions

Functions are relations between two sets: domain and range. The domain is the set of \(x\) values for which a function is defined and the range is the set of \(y\) values for which a function is defined. In a function, each element in the domain is mapped with exactly one element in the range.

Functions can be represented as algebraic equations, graphs, tables, or words. In an algebraic equation, the function is represented as an expression involving one or more variables. In a graph, the function is a set of ordered pairs graphed on the coordinate plane. In a tabular format, we show functions as a table of \(x\)’s and \(y\)’s. In words, we describe a function using a sentence or phrase that explains the relationship between the input and output.

Function Notation

We use function notation to represent a function with symbols and expressions. It is also easy to evaluate functions using functional notation. The most common function notation is \(f(x)\), read as “\(f\) of \(x\),” where \(f\) is the name of the function and \(x\) is the input variable. For example, the equation \(y = 2x - 7\) in function notation can be written as \(f(x) = 2x - 7\). This function takes an input value of \(x\), multiplies it by \(2\), and then subtracts \(7\) from the result.

Building Functions

We can build simple functions based on known relationships between variables. For example, take the formula for distance, \(D = r \cdot t\). Using the language of functions, we can say that distance traveled is a function of rate of speed and time. If the rate is a constant \(50\) mph, we can use the function notation, \(f(t)\), for distance traveled and write \(f(t) = 50t\). If we input a time value of \(6\) hours, the distance will be \(f(t) = 50 \cdot 6 = 300\) miles.

Piecewise Functions

Piecewise functions are functions that are defined by different rules for different intervals. They are written using function notation with different formulas specified for different parts. Take a look at the piecewise function below:

\[f(x) = \begin{cases} x^2 \, \, \text{if} \, \, x < 0 \\ 2x + 4 \, \, \text{if} \, \, 0 \leq x < 2 \\ x - 6 \, \, \text{if} \, \, x \geq 2 \end{cases}\]

This function has three different rules, one for each interval shown.

The first rule tellings that if \(x\) is in the interval of all \(x\) values less than zero, then \(f(x) = x^2\).

The second rule tells us that if \(x\) is in the interval of all \(x\) values greater than or equal to zero and less than two, then \(f(x) = 2x+4\).

The third rule tells us that if \(x\) is in the interval of all \(x\) values greater than or equal to two, then \(f(x) = x - 6\).

When graphing piecewise functions, it is important to graph each part separately and then connect them using open or closed circles, depending on whether the endpoints are included or excluded (\(<\) or \(\leq\) or \(>\) or \(\geq\)).

Geometric and Arithmetic Sequences

Number patterns are a sequence of numbers that follow a certain rule/pattern. They can follow arithmetic, geometric, or other types of patterns. Once the number pattern type is identified, you can use the appropriate formulas to find missing terms to extend the sequence.

For example, the sequence \(3, 5, 7, 9\) has a constant increase of \(2\) between terms, so we can predict that the next term will be \(9+2 = 11\). We can extend the pattern further by adding \(2\) to each term to get the next term.

Arithmetic Sequence

This type of sequence has a common difference between each term. To get a term of an arithmetic sequence, you have to add a fixed number, also called the common difference, to the previous term. The general formula for the \(n^{\text{th}}\) term of an arithmetic sequence is:

\[a_n = a_1 + (n-1)d\]

where \(a_n\) is the \(n^{\text{th}}\) term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.

Geometric Sequence

A geometric sequence is a sequence in which each term after the first is found by multiplying the preceding term by a constant number, also known as the common ratio. The formula for the \(n^{\text{th}}\) term of a geometric sequence is:

\[a_n = a_1 r^{n-1}\]

where \(a_n\) is the \(n^{\text{th}}\) term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

To determine missing terms of arithmetic and geometric sequences, we can use the formulas mentioned above. Simply plug in the given values and solve for the unknown.