Problem 5 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ y=\frac{1}{2} x+5 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = 2x - 10 \).

Step 1: Identify the Original Function

We have the function \( y = \frac{1}{2}x + 5 \). This is a linear function in the standard form \( y = mx + b \) where \( m = \frac{1}{2} \) and \( b = 5 \).

Step 2: Swap the Variables

To find the inverse of a function, start by swapping the dependent and independent variables. Set \( x = \frac{1}{2}y + 5 \).

Step 3: Solve for the New 'y'

Rearrange the equation \( x = \frac{1}{2}y + 5 \) to solve for \( y \):1. Subtract 5 from both sides: \( x - 5 = \frac{1}{2}y \).2. Multiply both sides by 2 to isolate \( y \): \( y = 2(x - 5) \).3. Simplify: \( y = 2x - 10 \).

Step 4: Replace 'y' with Inverse Notation

The equation \( y = 2x - 10 \) is the inverse of the original function. In inverse notation, this is \( f^{-1}(x) = 2x - 10 \).

Step 5: Graph the Original Function

Graph the original function \( y = \frac{1}{2}x + 5 \). Its graph is a straight line with a slope of \( \frac{1}{2} \) and a y-intercept at (0, 5).

Step 6: Graph the Inverse Function

Graph the inverse function, \( y = 2x - 10 \). This graph is also a straight line, with a slope of 2 and a y-intercept at (0, -10).

Step 7: Analyze the Graphs

Both graphs should intersect at the point where the line \( y = x \) would be present since they are inverses. This means they are symmetric around the line \( y = x \).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions in mathematics. They are represented in the form \( y = mx + b \), where \( m \) denotes the slope and \( b \) is the y-intercept of the function. These functions graph as straight lines on the coordinate plane. The slope \( m \) tells us how steep or flat the line is:

• If \( m > 0 \), the line slants upwards from left to right.
• If \( m < 0 \), the line slopes downwards from left to right.
• If \( m = 0 \), the line is perfectly horizontal.

The y-intercept \( b \) provides the point where the line crosses the y-axis (the vertical axis). In our given function \( y = \frac{1}{2}x + 5 \), the slope is \( \frac{1}{2} \), meaning for every unit increase in \( x \), \( y \) increases by \( 0.5 \) units. The y-intercept is \( 5 \), indicating the line crosses the y-axis at point (0, 5). Understanding these properties helps in sketching the graph accurately.

Graphing Functions

Graphing functions visually represents the relationship between variables. It involves plotting points on a coordinate grid defined by the x-axis (horizontal) and y-axis (vertical). To graph a linear function:

• Identify the y-intercept, \( b \), and place a point where the line meets the y-axis.
• Use the slope \( m \) to determine another point. With a slope \( m = \frac{1}{2} \), move right by 1 unit and up by 0.5 units.
• Draw a straight line through the points plotted.

For the inverse, repeat this process with \( f^{-1}(x) = 2x - 10 \). Start at the y-intercept \(-10\) and use the slope of 2 to find another point. Through graphing, not only do we see the line but also observe patterns like symmetry. Linear functions and their inverses always intersect at the line \( y = x \), showcasing their symmetrical nature. Graphing enhances understanding of function behavior and relationships.

Function Notation

Function notation is a way to express the relationship between elements of two sets using a formula. For example, \( f(x) = \frac{1}{2}x + 5 \) indicates that function \( f \) assigns each \( x \) a value \( y \), calculated by the given formula.Here are the essentials of function notation:

• The function name is often a single letter like \( f \), \( g \), or \( h \).
• \( f(x) \) means that \( f \) is a function of \( x \); "\( f \) of \( x \)."
• The expression shows how \( x \) transforms into \( y \) or the function's output.

The inverse function, denoted as \( f^{-1}(x) \), does the opposite. It finds the original \( x \) for any \( f(x) \). Here, the inverse \( f^{-1}(x) = 2x - 10 \) translates outputs of the original function back to their inputs. Function notation simplifies the process of working with and referencing functions, making it easy to communicate complex relationships clearly.