Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$f(x)=-5 x+2$$

Step-by-Step Solution

Answer

The function \( f(x) = -5x + 2 \) is one-to-one.

Step 1: Understand One-to-One Functions

A function is one-to-one if every distinct input gives a distinct output. In mathematical terms, if \( f(x_1) = f(x_2) \) implies that \( x_1 = x_2 \), then the function is one-to-one.

Step 2: Analyze the Given Function

The given function is \( f(x) = -5x + 2 \), which is a linear function with a non-zero slope. Linear functions with non-zero slopes are one-to-one because the slope ensures that different inputs will lead to different outputs.

Step 3: Algebraic Verification

To verify algebraically, assume \( f(x_1) = f(x_2) \). This gives us the equation \(-5x_1 + 2 = -5x_2 + 2\). Simplifying, we get \(-5x_1 = -5x_2\), which reduces to \(x_1 = x_2\). Hence, the function satisfies the condition for being one-to-one.

Step 4: Conclusion

Since \( f(x) = -5x + 2 \) satisfies the condition that \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \), it is a one-to-one function.

Key Concepts

Linear Function

A linear function is a type of function that creates a straight line when plotted on a graph, hence the name "linear." The general form of a linear function is \( f(x) = ax + b \), where:

• \( a \) is the slope, which determines the steepness or incline of the line.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

In the given exercise, the function \( f(x) = -5x + 2 \) is linear. Here, \( a = -5 \), indicating a slope of -5. This negative slope tells us that as \( x \) increases, \( f(x) \) decreases, producing a downward sloping line.
The y-intercept \( b = 2 \) means the line crosses the y-axis at the point (0, 2).
All linear functions with a non-zero slope are automatically one-to-one functions because they guarantee unique outputs for unique inputs.

Algebraic Verification

Algebraic verification is a method to determine the properties of a function by using algebraic manipulation. To verify if a function is one-to-one, we assume that the function's outputs are equal for two different inputs. Specifically, if \( f(x_1) = f(x_2) \), we should demonstrate that this implies \( x_1 = x_2 \).

Looking at the function \( f(x) = -5x + 2 \), assume \( f(x_1) = f(x_2) \). This leads to the equation:\[-5x_1 + 2 = -5x_2 + 2\]
By simplifying, subtract 2 from both sides to get:\[-5x_1 = -5x_2\]
Divide both sides by -5 to find:\[x_1 = x_2\]
Since we've shown that equal outputs imply equal inputs, the function is algebraically verified as one-to-one.

Function Properties

Functions can have various properties that define their behavior. Understanding these properties is key to analyzing and predicting how the function behaves under different conditions.- **One-to-One Function**: As detailed earlier, a function is labeled as one-to-one if each input results in a unique output. This can be a crucial property when trying to invert functions.- **Slope**: In linear functions like \( f(x) = -5x + 2 \), the slope is essential. A non-zero slope confirms a one-to-one relationship between inputs and outputs.- **Intercepts**: These are points where the function crosses the axes. For linear functions:

• **y-intercept**: The point where the function crosses the y-axis; for the given function, it is 2.
• **x-intercept**: The point where the function crosses the x-axis, calculated by setting \( f(x) = 0 \).

Understanding these properties helps students grasp the overall behavior and characteristics of functions, allowing them to apply this knowledge to diverse mathematical problems.