91Ó°ÊÓ

When working with a linear function like the one given in the exercise, finding the x-intercept is simple and straightforward. The x-intercept is the point where the graph of the function crosses the x-axis. This is the point where the output, also known as the function value, is zero. In mathematical terms, it's where \( f(x) = 0 \).
To find this x-intercept for the function \( f(x) = 3x - 6 \), you set the equation to zero and solve for \( x \). This means you solve \( 0 = 3x - 6 \), which leads to \( x = 2 \) after solving the equation.
The coordinates of the x-intercept are therefore \( (2, 0) \), because it's a point on the graph where the y-value, representing the output of the function, is zero.

The y-intercept of a linear function is equally easy to understand as the x-intercept. It's the point where the function crosses the y-axis, which happens when the x-value is zero. In practical terms, you find this by plugging in zero for \( x \) in the equation and solving for \( f(x) \), or \( y \).
For the function \( f(x) = 3x - 6 \), substitute \( x = 0 \) into the equation, resulting in \( f(0) = -6 \). Thus, the y-intercept is at the point \( (0, -6) \).

• This means the graph touches the y-axis at the point where the y-value is -6.

Linear functions are defined everywhere on the real number line, which means their domain is all real numbers. For the function \( f(x) = 3x - 6 \), there are no restrictions on the values that \( x \) can take. Hence, the domain is represented as \(( -\infty, \infty )\).

• The function has no breaks, holes, or gaps within the real numbers.

Similarly, the range of a linear function is all real numbers since the line extends infinitely in both upward and downward directions on the graph. Therefore, the range of \( f(x) = 3x - 6 \) is also \(( -\infty, \infty )\).
In simple terms, for every x-value, there's a corresponding y-value, meaning the line continues forever along the y-axis.

The slope is a crucial aspect of the linear function, indicating how steep the line is and in which direction it inclines. A positive slope means the line rises as it goes from left to right, while a negative slope means the line falls.For the function \( f(x) = 3x - 6 \), it's in the form \( y = mx + b \), where \( m \) is the slope. Here, \( m = 3 \), indicating that for every one unit increase in \( x \), \( y \) increases by 3 units.

• This is described as a slope of 3, a moderate incline showing how rapidly y increases with x.

Understanding the slope helps in predicting how any changes in \( x \) will affect the outcome or \( y \) value of the function.