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Magazine Chapter 12: Problem 25
Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=2 x-3 $$
Short Answer
Expert verified
The inverse function is \( f^{-1}(x) = \frac{x + 3}{2} \).
Step by step solution
01
Understand the function
The function given is \( f(x) = 2x - 3 \). This is a linear function and it is one-to-one, meaning it has a unique inverse. The goal is to find this inverse function.
02
Set up the equation for inversion
To find the inverse of the function \( f(x) = 2x - 3 \), we begin by replacing \( f(x) \) with \( y \). This gives us the equation \( y = 2x - 3 \).
03
Solve for x
To find the inverse, solve the equation for \( x \) in terms of \( y \). Start by adding 3 to both sides: \( y + 3 = 2x \). Then, divide both sides by 2: \( x = \frac{y + 3}{2} \).
04
Express the inverse function
The expression \( x = \frac{y + 3}{2} \) is our inverse function when expressed as \( x(y) \). To express it in standard form as \( f^{-1}(x) \), swap the \( x \) and \( y \), giving us the inverse function: \( f^{-1}(x) = \frac{x + 3}{2} \).
05
Graphing the functions
To graph both \( f(x) \) and \( f^{-1}(x) \), plot the function \( f(x) = 2x - 3 \) as a straight line. The slope is 2 and the y-intercept is -3. For the inverse \( f^{-1}(x) = \frac{x + 3}{2} \), the slope is 0.5 and the y-intercept is 1.5. Remember that both graphs should be mirror images across the line \( y = x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
One-to-One Function
A one-to-one function is a special type of function that has a unique output for each unique input. In simpler terms, every x-value leads to one and only one y-value. Similarly, every y-value comes from a unique x-value. This uniqueness is crucial for a function to have an inverse.
If a function is not one-to-one, its inverse wouldn't truly exist. This is because an inverse function requires reversing the process: taking y-values back to x-values uniquely. If two different x-values produced the same y-value, you wouldn't know which to choose when going backward.
If a function is not one-to-one, its inverse wouldn't truly exist. This is because an inverse function requires reversing the process: taking y-values back to x-values uniquely. If two different x-values produced the same y-value, you wouldn't know which to choose when going backward.
- Does not have repeating y-values for different x-values.
- Passes the horizontal line test: Any horizontal line should intersect the graph of the function at most once.
- Has an inverse function that is also a one-to-one function.
Linear Function
A linear function is a function whose graph is a straight line. The general formula for a linear function is:
\[ f(x) = mx + b \]where:
Linear functions are always one-to-one unless the slope is zero (creating a horizontal line). This means they generally have inverses, like the function in the exercise, \( f(x) = 2x - 3 \). Here, the slope \( m \) is 2, indicating a rising line as you move from left to right.
\[ f(x) = mx + b \]where:
- \( m \) represents the slope of the line, showing how steep it is.
- \( b \) is the y-intercept, where the line crosses the y-axis.
Linear functions are always one-to-one unless the slope is zero (creating a horizontal line). This means they generally have inverses, like the function in the exercise, \( f(x) = 2x - 3 \). Here, the slope \( m \) is 2, indicating a rising line as you move from left to right.
Graphing Functions
Graphing functions is about representing functions visually on a coordinate plane. This helps us understand their behavior, like where they increase, decrease, or meet other lines.
For the function \( f(x) = 2x - 3 \):
For the function \( f(x) = 2x - 3 \):
- Identify the slope (2) and y-intercept (-3).
- Start by plotting the y-intercept: a point at (0, -3) on the y-axis.
- From this point, use the slope to find another point: go up 2 units and one unit to the right.
- The slope is 0.5, and the y-intercept is 1.5.
- Plot the point (0, 1.5) and use the slope to find another: up 0.5 units and one unit right.
Slope and Intercept
The slope and intercept are key in understanding how a linear function behaves.
The slope is a measure of steepness and direction:
The slope is a measure of steepness and direction:
- If the slope is positive, the function rises as it moves from left to right.
- If the slope is negative, the function falls.
- A larger absolute value means a steeper line, smaller means flatter.
- It tells you the starting point of the function when x equals zero.
- For the inverse function in our case, the y-intercept is 1.5, different from the original function's -3.
