An equation that defines y as a function f of x is given....

An equation that defines $y$ as a function $f$ of $x$ is given. (a) Solve for $y$ in terms of $x$, and write each equation using function notation $f(x) .$ (b) Find $f(3) .$ See Example 6.
$$
-2 x+5 y=9
$$

Key Concepts

Linear Equations

Linear equations are algebraic statements that represent straight lines when graphed in the coordinate plane. They typically involve variables raised only to the first power and can be rearranged into various forms, such as the slope-intercept form, which makes it easy to identify important properties like slope and intercepts. Understanding linear equations is fundamental in algebra and various applications across mathematics.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions using arithmetic operations and properties of equality. This is particularly important for isolating variables in an equation, allowing one to express one variable in terms of others, which is a key step in solving equations and converting them into more useful forms.

Function Notation

Function notation is a concise way to represent relationships between two variables, typically written as f(x) to denote the output of the function when x is the input. This notation emphasizes that the relationship is a function, meaning each input x corresponds to exactly one output f(x). Function notation is widely used in mathematics to simplify the expression and analysis of relationships between quantities.

Evaluating Functions

Evaluating a function means substituting a specific value into the function and simplifying to obtain the corresponding output. This process is crucial for understanding how a function behaves at particular points and for analyzing and graphing the function.

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