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The x-intercept of a line is the point where the line crosses the x-axis. At this intercept, the value of the function, or the output, is zero. This means that we need to find the value of \( x \) when \( f(x) = 0 \). For example, in the equation \( f(x) = -x + 4 \), we solve for \( x \) by setting the equation to zero:

• Replace \( f(x) \) with 0: \( 0 = -x + 4 \)
• Rearrange terms: \( x = 4 \)

The x-intercept is therefore the point \((4, 0)\). It's where the line meets the x-axis at \( x = 4 \).

The y-intercept is the point where the line crosses the y-axis. This occurs when the value of \( x \) is zero. To find the y-intercept in the equation \( f(x) = -x + 4 \), substitute \( x = 0 \) into the function:

• Substitute 0 for \( x \): \( f(0) = -0 + 4 \)
• Simplify the equation: \( f(0) = 4 \)

Thus, the y-intercept is \((0, 4)\). This is where the line meets the y-axis at \( y = 4 \).

In the context of linear equations, the domain and range refer to the set of possible input (\( x \)) and output (\( y \)) values. **Domain**: The domain of a linear function is all real numbers, represented by \((-\infty, \infty)\). This is because a straight line will extend infinitely to the left and right on a graph. **Range**: Similarly, the range is all real numbers, \((-\infty, \infty)\), because a straight line will also continue infinitely up and down on a graph. In summary, both the domain and range for the function \( f(x) = -x + 4 \) include all real numbers, showcasing the infinite nature of linear functions. Linear equations will always have these properties as long as they do not have constraints.

The slope of a line describes its steepness and direction. It's often denoted as \( m \) in the slope-intercept form. The slope shows how much \( y \) changes for a change in \( x \).For the line \( f(x) = -x + 4 \) in the form \( y = mx + b \), \( m = -1 \). This slope value indicates:

• The line goes downwards as it moves from left to right, because the slope is negative.
• For every 1 unit increase in \( x \), \( y \) decreases by 1 unit.

Understanding slope is essential for recognizing how lines behave and interact with graphs, as well as interpreting the real-world meaning of relationships in graphs.

The slope-intercept form is a common way to express linear equations. It is written as \( y = mx + b \), where:

• \( m \) represents the slope of the line.
• \( b \) represents the y-intercept, the point where the line crosses the y-axis.

For the equation \( f(x) = -x + 4 \), this can be directly compared with \( y = mx + b \) to see:

• \( m = -1 \), the slope.
• \( b = 4 \), the y-intercept.

This form is extremely useful because it provides immediate insight into two key features of the line: how steep it is, and where it starts on the y-axis. By simply looking at an equation in this form, you can easily sketch the graph of the line.