Problem 7 - Algebra for College Students

Question

Determine whether the function \(f\) is one-to-one. $$ f(x)=5 x+4 $$

Step-by-Step Solution

Answer

The function is one-to-one.

Step 1: Understanding One-to-One Functions

A function is considered one-to-one if different inputs produce different outputs. This means that for any two distinct values, say \(x_1\) and \(x_2\), if \(x_1 eq x_2\), then \(f(x_1) eq f(x_2)\).

Step 2: Testing with Equation Setup

To determine if the function is one-to-one, set \(f(x_1) = f(x_2)\) and see if this leads to \(x_1 = x_2\). For the given function \(f(x) = 5x + 4\), set \(5x_1 + 4 = 5x_2 + 4\).

Step 3: Simplifying the Equation

Subtract 4 from both sides of the equation \(5x_1 + 4 = 5x_2 + 4\), which gives \(5x_1 = 5x_2\).

Step 4: Solving for Variables

Divide both sides of the equation \(5x_1 = 5x_2\) by 5, yielding \(x_1 = x_2\). Since we started with \(f(x_1) = f(x_2)\) and ended with \(x_1 = x_2\), the function is one-to-one.

Key Concepts

Function Properties

Functions play a crucial role in mathematics as they establish relationships between different sets of numbers. A function can have several properties that help describe these relationships. One key property is being a "one-to-one" function. In a one-to-one function, every input value maps to a unique output value. This means no two input values share the same output.

• Every input has a distinct output.
• No repeating output values for different inputs.

Identifying whether a function is one-to-one can be crucial when working with inverses, as only one-to-one functions have inverses that are also functions. Checking a function's "one-to-one" property can be done by examining its graph; if it passes the horizontal line test (no horizontal line cuts the graph more than once), it's one-to-one.
Understanding function properties helps in working with complex mathematical problems and ensures the correct application of functions in various scenarios.

Linear Functions

Linear functions are among the simplest types of functions in mathematics. They create straight lines when graphed and have a standard form:
\[ f(x) = mx + b \] Here,

• \( m \) is the slope, determining the line's steepness.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

Linear functions are always one-to-one, provided the slope \( m \) is not equal to zero. This is because any point on the line has a unique y-value for any x-value, except when the line is horizontal (\( m = 0 \)).
The simplicity of linear functions makes them a favorite for modeling in various fields such as business and economics. They provide a straightforward representation of constant change.

Function Verification

Verifying that a function possesses certain properties, like being one-to-one, involves some straightforward testing. Let's apply this to the linear function from the original exercise:
\[ f(x) = 5x + 4 \] To verify it's one-to-one, we set \( f(x_1) = f(x_2) \) and check the equality of inputs:

• Given \( f(x_1) = 5x_1 + 4 \) and \( f(x_2) = 5x_2 + 4 \), set \( 5x_1 + 4 = 5x_2 + 4 \).
• Subtract 4 from both sides: \( 5x_1 = 5x_2 \).
• Divide by 5: \( x_1 = x_2 \).

Since we end up with \( x_1 = x_2 \), the function is verified as one-to-one.
This process of verification is important, especially when dealing with transformations or trying to establish inverses, ensuring that each step mathematically confirms the properties in question.