Functions and Linear Equations

Ordered Pairs and Solutions to Function Equations

Understanding whether an ordered pair is a solution to a function equation is fundamental in algebra. An ordered pair (x, y) is a solution to a function equation if, when x is substituted into the equation, the resulting y value matches the second component of the pair.

• Definition: A function equation relates x (input) to y (output) such that each input has exactly one output.

• Testing a Solution: Substitute the x-value from the ordered pair into the function equation. If the resulting y-value equals the y-value in the ordered pair, it is a solution.

• Finding a Solution: To find an ordered pair solution, choose a value for x, substitute it into the equation, and solve for y.

• Example: Given the function , is (1, 5) a solution?

  • Substitute x = 1:

  • Since y = 5, (1, 5) is a solution.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. Determining the domain often involves identifying values that would cause division by zero or taking the square root of a negative number.

• Notation: Domains are typically written in interval notation or set notation.

• Steps to Find the Domain:

  1. Identify any restrictions (e.g., denominators cannot be zero, radicands of even roots must be non-negative).

  2. Express the set of all valid x-values using appropriate notation.

• Example: For , the denominator cannot be zero, so . The domain is .

Slope of a Linear Function in Real-World Applications

The slope of a linear function measures the rate of change between two variables. In real-world contexts, the slope often represents a rate such as speed, cost per item, or change over time.

• Definition: The slope (m) of a line is the ratio of the change in y to the change in x:

• Interpretation: In applications, the slope tells how much y changes for each unit increase in x.

• Example: If a taxi charges $3 per mile, the slope of the cost function is 3, meaning the cost increases by $3 for each additional mile.

Graphing a Linear Equation

Graphing linear equations is a key skill in algebra. A linear equation graphs as a straight line, and can be represented in several forms, such as slope-intercept or standard form.

• Slope-Intercept Form: , where m is the slope and b is the y-intercept.

• Standard Form:

• Steps to Graph:

  1. Identify the y-intercept (b) and plot it on the y-axis.

  2. Use the slope (m) to find another point: rise over run.

  3. Draw a straight line through the points.

• Example: For , the y-intercept is -1, and the slope is 2. Plot (0, -1), then from there, go up 2 units and right 1 unit to plot a second point.

Writing the Equation of a Line

Writing the equation of a line involves using information such as a point and a slope, or two points, to construct the equation in a desired form.

• Point-Slope Form: , where (x1, y1) is a point on the line and m is the slope.

• Slope-Intercept Form:

• Standard Form:

• Steps:

  1. If given two points, first find the slope:

  2. Use point-slope form with one of the points.

  3. Rearrange to slope-intercept or standard form as needed.

• Example: Find the equation of the line through (2, 3) with slope 4.

  • Point-slope form:

  • Simplify: