Problem 51 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Sketch the graph of \(f\) (b) Use the graph of \(f\) to sketch the graph of \(f^{-1} .\) (c) Find \(f^{-1} .\) \(f(x)=3 x-6\)

Step-by-Step Solution

Answer

(a) Linear graph with slope 3, y-intercept -6. (b) Reflect over line \(y=x\). (c) \(f^{-1}(x) = \frac{x}{3} + 2\).

Step 1: Sketch the Graph of f

The given function is \(f(x) = 3x - 6\). This is a linear function in the slope-intercept form \(y = mx + b\), where \(m = 3\) (the slope) and \(b = -6\) (the y-intercept). To sketch the graph of \(f\), plot the y-intercept, which is the point \((0, -6)\), then use the slope to determine another point. The slope of 3 means for every 1 unit you move to the right, you move up 3 units. Plot another point using this slope, such as \( (1, -3) \), and draw a line through these points.

Step 2: Sketch the Graph of f^{-1}

To find the graph of \(f^{-1}\), we reflect the graph of \(f\) across the line \(y = x\). This involves swapping the x- and y-coordinates of each point on the graph of \(f\). For example, the points \((0, -6)\) and \((1, -3)\) on \(f\) become \((-6, 0)\) and \((-3, 1)\) on \(f^{-1}\). Plot these reflected points and draw a line through them to get the graph of \(f^{-1}\).

Step 3: Find f^{-1} algebraically

To find \(f^{-1}\) algebraically, we start with the equation \(y = 3x - 6\) and solve for \(x\) in terms of \(y\). Switch \(x\) and \(y\) to get \(x = 3y - 6\). Solve for \(y\) to get \(3y = x + 6\), and finally, \(y = \frac{x}{3} + 2\). Thus, the inverse function is \(f^{-1}(x) = \frac{x}{3} + 2\).

Key Concepts

Linear Function

A linear function is an algebraic equation that forms a straight line when graphed on a coordinate plane. It's one of the simplest types of functions and is widely used in various fields such as economics, physics, and everyday problems.
To identify a linear function, look for the general form:

• Slope-intercept form: \(y = mx + b\)

In this form, \(m\) represents the slope, which tells us how steep the line is, and \(b\) is the y-intercept, the point where the line crosses the y-axis.
Linear functions are characterized by their constant rate of change. That means the slope is the same anywhere on the line. For the function \(f(x) = 3x - 6\), the slope \(m = 3\) means for every increase of 1 in \(x\), \(f(x)\) increases by 3. The y-intercept \(b = -6\) indicates the line crosses the y-axis at \((0, -6)\).
By understanding these components, you can quickly graph a linear function and predict its behavior.

Graph Sketching

Graph sketching involves plotting points on a coordinate plane to visually represent a function. It helps in understanding the function's behavior and making predictions.
To sketch the graph of a linear function like \(f(x) = 3x - 6\), follow these steps:

• Plot the y-intercept: Start by marking the point where the function crosses the y-axis, which for \(f(x)\) is at \((0, -6)\).
• Use the slope: From the y-intercept, apply the slope to find another point. With a slope of 3, move 1 unit right (positive direction) and 3 units up from \((0, -6)\). This gives you the point \((1, -3)\).

Drawing a line through these points gives the graph of \(f(x)\).
Reflecting this graph over the line \(y = x\) helps sketch the inverse function's graph. Swap x and y coordinates of all points on the original graph to reflect, making new points like \((-6, 0)\) and \((-3, 1)\) for the inverse \(f^{-1}(x)\).
Graph sketching not only provides a visual representation but also reinforces the understanding of algebraic manipulations.

Slope-Intercept Form

The slope-intercept form, denoted as \(y = mx + b\), is a powerful tool in graphing linear functions and quickly understanding their characteristics.

• Slope (\(m\)): Indicates the direction and steepness of the line. A positive slope means the line rises as x increases, while a negative slope means it falls.
• Y-intercept (\(b\)): Shows where the line meets the y-axis. It tells you the value of \(y\) when \(x = 0\).

For \(f(x) = 3x - 6\), the slope \(m = 3\) suggests a rising line, moving upwards 3 units for each 1 unit rightward. The y-intercept \(b = -6\) shows the line starts at \((0, -6)\), below the x-axis.
Understanding the slope-intercept form enables you to instantly position and orient a line on a graph, making it easier to interpret equations and solve real-world problems. It's also crucial in finding inverse functions, as seen in the step to solve for \(f^{-1}(x) = \frac{x}{3} + 2\), where the new slope and y-intercept rearrange the line to reflect across \(y = x\).
Mastering this form is essential for efficiently working with linear equations in mathematics.