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The x-intercept of a function is the point where the graph of the function crosses the x-axis. This is where the output, or the y-value, is zero. For rational functions, figuring out the x-intercept involves setting the function equal to zero and solving for x.

For example, let's consider the rational function given by the formula:

• \( s(x) = \frac{3x}{x-5} \).

To find the x-intercept, we set \( s(x) = 0 \) and this gives us the equation:

• \( \frac{3x}{x-5} = 0 \).

In a fraction, the only way for it to equal zero is if the numerator is zero while the denominator is not zero. Hence, solving \( 3x = 0 \) results in \( x = 0 \). This means the graph intersects with the x-axis at the point \( (0, 0) \), known as the x-intercept.

It's vital to check that the denominator, \( x - 5 \), is never zero at this intercept, ensuring that the expression is defined. Remember, division by zero is not allowed!

The y-intercept of a function is the point where the graph crosses the y-axis. At this point, the x-value is always zero. Finding the y-intercept involves substituting zero for x in the function and evaluating the result. Let's look again at our function:

• \( s(x) = \frac{3x}{x-5} \).

To find the y-intercept, we substitute \( x = 0 \) into the function. The equation becomes:

• \( s(0) = \frac{3 \times 0}{0 - 5} = \frac{0}{-5} = 0 \).

This calculation shows that the y-intercept is also at the origin, the point \( (0, 0) \). Both the x- and y-intercepts coincidentally occur at the same point in this specific case.

Locating the y-intercept is essential because it provides a clear point of reference when graphing the function and aids in understanding the function's initial value at the origin.

Solving equations is a fundamental skill in mathematics that involves finding the values of variables that make the equation true. With rational functions, such as \( s(x) = \frac{3x}{x-5} \), solving for intercepts involves creating simpler equations by setting the function equal to zero or inputting specific values.

When solving for an x-intercept, the key step is to equate the rational function to zero and focus on making the numerator zero, assuming the denominator is nonzero. For our specific function:

• \( \frac{3x}{x-5} = 0 \) simplifies to \( 3x = 0 \).

This process makes solving straightforward as we only deal with a linear equation.

For the y-intercept, recognizing that \( x = 0 \) simplifies the rational function and leads directly to the result after evaluation.

• Every step relies on understanding that all mathematical operations must adhere to existing rules, like avoiding division by zero.

Overcoming these challenges allows for a precise determination of the intercepts, enlightening broader implications about the function's behavior.