Problem 39 - Calculus with Concepts in Calculus

Question

Determine the range of the function. $$ f(x)=3 x-2 \text { for } x<4 $$

Step-by-Step Solution

Answer

The range of the function is \((- underf enity, 10)\).

Step 1: Understand the function

The function given is linear, specifically \( f(x) = 3x - 2 \). It is defined for values \( x < 4 \). Our task is to find all possible values that \( f(x) \) can take when \( x \) is less than 4.

Step 2: Use the inequality condition to find the domain

The inequality \( x < 4 \) means the domain of \( f \) includes all real numbers that are less than 4. Therefore, the domain is the interval \((- underf enity, 4)\).

Step 3: Find the value of the function at the boundary of the domain

Although \( x = 4 \) is not included because \( x \) must be less than 4, calculating \( f(4) \) helps understand its behavior near this boundary. Compute:\( f(4) = 3(4) - 2 = 12 - 2 = 10 \).

Step 4: Determine the behavior of the function as \( x \) approaches boundary conditions in the domain

Since \( f(x) \) is a linear equation with a positive slope (3), it's increasing. This implies that as \( x \) approaches 4 from the left, \( f(x) \) increases toward but does not include 10.

Step 5: Determine the range of the function

As \( x \) moves to very negative values (approaching \(- underf enity \)), \( 3x - 2 \) also decreases, approaching \(- underf enity \). Thus, the range of \( f(x) \) includes all values less than 10, defined as \((- underf enity, 10)\).

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions to work with. They can be easily identified as functions of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

• The graph of a linear function is a straight line.
• The coefficient \( a \) is the slope, which dictates the slant of the line; if \( a > 0 \), the line rises, while if \( a < 0 \), it falls.
• The constant \( b \) determines the y-intercept, where the line crosses the y-axis when \( x = 0 \).

When determining a linear function, such as \( f(x) = 3x - 2 \), its behavior and attributes are dictated by these constants.
The slope \( a = 3 \) indicates that the function increases by 3 units for every 1-unit increase in \( x \). This linear growth is why the range is continually increasing or decreasing linearly.

Domain and Range

Understanding the domain and range of a function is crucial for analyzing its behavior. The domain refers to all possible input values (\( x \) values), while the range includes all possible output values (\( f(x) \) values).

• The domain is usually the set of all real numbers for simple functions, but it can be restricted by conditions such as those specified in the problem: \( x < 4 \).
• For the given function \( f(x) = 3x - 2 \), this restriction creates a domain of \((-\infty, 4)\).

The range of a function depends on its domain and the function's nature.
Since this is a linear function, the output (\( f(x) \)) will depend on the input \( x \), specifically how it behaves as \( x \) approaches certain key values.
In this case, as \( x \) nears 4 from the left, \( f(x) \) comes close to 10, setting the upper boundary for the range.

Inequality Conditions

Inequality conditions help define specific intervals for which functions are evaluated.

• They constrain the input or output values, as seen in our example where \( x < 4 \) delineates the domain of the function.
• These conditions shape the resultant range, as the values \( f(x) \) can take are directly influenced by where \( x \) lies in relation to these inequalities.

For \( f(x) = 3x - 2 \), the inequality \( x < 4 \) means examining behavior as \( x \) gets closer to this boundary without surpassing it.
Such behavior shows how, upon approaching 4, f(x) approaches but never reaches 10, defining its range as \((-\infty, 10)\).
This concept highlights why understanding inequalities is important in solving and interpreting mathematical problems pertaining to functions.