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The x-intercept of a linear function is a crucial point where the graph of the function cuts across the x-axis. To find the x-intercept of a function like \( f(x) = \frac{4}{3}x - 3 \), you set the function equal to zero and solve for \( x \). This is because, at the x-intercept, the value of \( y \) (or \( f(x) \)) is zero. So for our function, we set up the equation:\[\frac{4}{3}x - 3 = 0\]By adding 3 to both sides and then multiplying by \( \frac{3}{4} \), we find that \( x = \frac{9}{4} \). This means the line crosses the x-axis at \( x = \frac{9}{4} \).It's important to note:

• The x-intercept provides the point at which the function is equal to zero.
• This is valuable for finding solutions to equations and understanding the behavior of graphs.

The y-intercept is the spot where the function intersects the y-axis. Here, the value of \( x \) is zero. To determine the y-intercept, plug \( x = 0 \) into the equation, resulting in \( f(0) = \frac{4}{3}(0) - 3 = -3 \). Therefore, the y-intercept of the function is \( y = -3 \).The y-intercept is easy to find as it can usually be read directly from the equation, especially if it's in the form \( y = mx + b \) where \( b \) is the y-intercept.Consider these:

• The y-intercept indicates where the graph of a function will cross the y-axis.
• It provides a way to quickly graph linear functions.

Both the domain and range of a linear function describe the extent of the input and output values of the function. For our function \( f(x) = \frac{4}{3}x - 3 \), these concepts are straightforward.

Domain

The domain of a linear function encompasses all possible \( x \)-values it can accept. Linear functions, like the one we are examining, are defined for every real number, which means the domain is \( (-\infty, \infty) \).

Range

The range comprises all resulting \( y \)-values when the function is applied to its domain. For a linear function, the range is also \( (-\infty, \infty) \), as the line can extend indefinitely up and down on the graph.In summary:

• Linear functions generally have domains and ranges of all real numbers, unless restricted by context or additional conditions.
• Their graphs are straight lines that extend infinitely in both directions.

The slope of a linear function refers to the steepness and direction of the line. In the equation \( f(x) = \frac{4}{3}x - 3 \), the slope is represented by the coefficient of \( x \), which is \( \frac{4}{3} \). This shows that for every 3 units moved horizontally (right), the line rises by 4 units vertically.Understanding slope is critical because:

• If the slope is positive, the line ascends from left to right, indicating an increasing function.
• If the slope is negative, the line descends from left to right, indicating a decreasing function.
• A larger absolute value of the slope signifies a steeper line.

In practical terms, the slope can also represent rates of change and is foundational in calculus and physics to describe motion and trends.