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Section 5.1 Summary of Notes

Subsection 5.1.1 Function Notation

Table 5.1. Evaluating and Solving with Functions
Format of Function Evaluate \(f(2)\) (2 is the input) Solve \(f(x)=2\) (2 is the output)

Formula: The variable inside parentheses (usually \(x\) or \(t\)) is a placeholder for the input. Both \(f(x)\) and \(y\) represent the output.

Replace all of the input variables (\(x\) or \(t\)) in the formula with \(2\text{.}\)

Set the formula equal to 2 and solve for the input variable (\(x\) or \(t\))

Table: If the table is in rows, the row with the \(x\) at the front is the input row and the row with the \(f(x)\) at the front is the output row. If the table is in columns, the column with the \(x\) on top is the input column and the column with the \(f(x)\) on top is the output column. If there is more than one function, each one will have its own output row/column, but they share an input row/column.

Find 2 in the input row\column, then go to the corresponding location in the output row\column.

Find 2 in the output row\column, then go to the corresponding location in the input row\column.

Graph: The \(x\)-coordinates are the inputs and the \(y\)-coordinates are the outputs. If the graph has multiple parts, a filled-in endpoint ("closed point") is included but an endpoint that is not filled in ("open point") is not included.

Find \(x=2\) on the \(x\)-axis, then look up and down for the \(y\)-value where it crosses the graph.

\(y=2\) on the \(y\)-axis, then look left and right for the \(x\)-value(s) where it crosses the graph.

If the input has variables in it, replace all of the \(x\)'s in the function with the entire expression inside the parentheses. See Example 2.4 for an example.

If you have an expression with more than one function or numbers outside of the function, first evaluate the function(s) on its own, then plug that output into the expression. See Checkpoint 2.25 for an example.

For piecewise functions, first use the inequalities on the right to determine which formula to use, then plug the input into only the correct formula. See Example 2.10 for an example.

Subsection 5.1.2 Composition of Functions

Evaluate the inside function first. Its output becomes the input for the next function.

If there is no number given for the input, the entire formula for the inside function is the input to the outside function.

Subsection 5.1.3 Inverse Functions

The notation \(f^{-1}(x)\) represents the inverse function and is read out loud as "f inverse of x".

The input to the inverse is the output to the original. So, \(f^{-1}(2)\) is the same as \(f(x)=2\text{.}\)

To find the formula for the inverse function:

  1. Write \(y=f(x)\)

  2. Swap \(x\) and \(y\)

  3. Solve for \(y\)