Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

How do the \(x\) - and \(y\) -intercepts of \(f(x)\) and \(g(x)\) affect the intercepts of \(\frac{f(x)}{g(x)}\) and the places where \(\frac{f(x)}{g(x)}\) is unde ned?

Step-by-Step Solution

Answer

\(f(x)/g(x)\) will have \(x\)-intercepts where \(f(x)\) is zero, provided \(g(x)\) is not zero at these points. If \(g(x)\) is zero at these points, these points will be undefined. The \(x\)-intercepts of \(f(x)/g(x)\) are where \(f(x)\) is zero and \(g(x)\) is not zero. The function \(f(x)/g(x)\) is undefined where \(g(x) = 0\), but \(f(x) \neq 0\).

Step 1: Understand the Intercept Concept

The \(x\)-intercept of a function is the point or points where the function intersects with the \(x\)-axis and the function output \(f(x)\) is zero. The \(y\)-intercept is the point where the function intersects the \(y\)-axis, typically when \(x\) is zero. For a defined function \(y = f(x) / g(x)\), \(f(x)\) must be zero to have \(x\)-intercepts and \(g(x)\) must not be zero to avoid undefined points.

Step 2: Identify the Intercepts of the Resulting Function

When \(f(x)\) and \(g(x)\) have the same \(x\)-intercepts, the resulting function \(f(x)/g(x)\) will also have these as its own \(x\)-intercepts. This is because if \(f(x) = 0\) then \(f(x)/g(x) = 0/g(x) = 0\), provided that \(g(x)\) is not zero at these points. If \(g(x)\) is zero at these points, these become points of discontinuity.

Step 3: Understanding the Undefined Points

For a function \(f(x) / g(x)\), points where \(g(x) = 0\), but \(f(x) \neq 0\) are points of discontinuity or undefined points. This is because division by zero is undefined in mathematical operations. Therefore, the \(x\)-values at these points do not belong to the domain of the function \(f(x)/g(x)\).

Key Concepts

X-intercept

To understand where a rational function meets the x-axis, consider the x-intercept—part of solving equations and visualizing graphs. An x-intercept occurs where a function's output, f(x), equals zero. This means the graph of f(x) crosses the x-axis, providing a key characteristic of the function's behavior. When dealing with a fraction like f(x)/g(x), finding this intercept requires that we set the numerator, f(x), to zero. However, the denominator, g(x), must not be zero to ensure the fraction is defined at that point.

Imagine the graph of a function gently touching or crossing the x-axis. Those points at which it makes contact are the x-intercepts. For the rational function f(x)/g(x), these points are ones where f(x)=0, but only if g(x) is not zero. If the numerator and denominator share common factors that can be cancelled out, the function may have removable discontinuities or 'holes' rather than intercepts, which is essential to note for a complete understanding.

Y-intercept

While the x-intercept tells us where a function intersects the x-axis, the y-intercept reveals the crossing point on the y-axis. This happens when the input x is zero. Most functions exhibit their y-intercept as one distinct point, where f(0) is the output for x=0. For the quotient of functions, f(x)/g(x), calculate the y-intercept by evaluating f(0)/g(0), presuming g(0) is not equal to zero.

Finding the y-intercept is typically straightforward: plug zero into the function and simplify. In the context of a rational function, as long as the denominator is non-zero when x is zero, the y-intercept will be the quotient of the numerator and denominator evaluated at x=0. This single point is the only y-intercept a function can have, making it a simple but significant feature on the graph.

Undefined Points in Functions

A key challenge when dealing with rational functions like f(x)/g(x) is identifying undefined points. These occur entirely due to the denominator g(x). If g(x) equals zero at any point, the function becomes undefined for that input x, causing a break in the graph, often termed as 'discontinuity'.

An intuitive way to think about undefined points is by imagining them as holes or gaps in the graph. This means the function cannot produce a real number output for those particular x-values. Algebraically, they are found by setting the denominator equal to zero and solving. Undefined points are crucial when sketching function graphs, as they let you differentiate between continuous sections of the graph and places where it jumps or halts.

Division by Zero in Functions

Division by zero is a mathematical 'no-no'. Whenever a function calls for division by zero, it leads to an undefined situation. In graphical terms, the function might have vertical asymptotes—or lines that the graph approaches but never touches—at these points.

In the context of rational functions, when g(x) equals zero at any point where f(x) doesn't, the function f(x)/g(x) cannot have a real value there. These specific points are not part of the domain of the function, and they cannot be graphed on the regular coordinate plane. Understanding the restrictions division by zero imposes is vital to grasp the overall behavior and limits of functions, as it affects both the domain and the graphical representation.