Problem 14 - The Calculus with Analytic Geometry

Question

Determine the intervals on which the given function is continuous. $$ g(y)=\frac{4}{y+1} $$

Step-by-Step Solution

Answer

The function \( g(y) \) is continuous on \( (-\infty, -1) \cup (-1, \infty) \).

Step 1: Identify the function

The given function is \( g(y) = \frac{4}{y+1} \).

Step 2: Determine where the denominator is zero

A rational function is continuous everywhere except where its denominator is zero. Set the denominator equal to zero and solve for \( y \): \[ y+1 = 0 \] \[ y = -1 \]

Step 3: Define the intervals of continuity

The function \( g(y) \) is continuous for all \( y \) except where it causes division by zero. Thus, \( g(y) \) is continuous on the intervals \( (-\infty, -1) \cup (-1, \infty) \).

Key Concepts

rational functions

Rational functions are a type of function that can be expressed as the ratio of two polynomials. For example, in the function \( g(y) = \frac{4}{y+1} \), the numerator is 4 and the denominator is \( y+1 \). These functions are helpful in many areas of math and science because they can describe a wide variety of behaviors.

However, understanding rational functions involves dealing with their unique characteristics, such as vertical asymptotes where the function is undefined. We find these by setting the denominator equal to zero and solving for the variable. This will help us understand where the function can or cannot be continuous.

intervals of continuity

Continuity of a function means that the function can be drawn without lifting the pencil from the paper. For rational functions like \( g(y) = \frac{4}{y+1} \), it is essential to identify where the function is not continuous. To do this, look at the intervals on the real number line where the function remains defined and does not involve division by zero.

In this case, the function is not defined at \( y = -1 \) because it causes the denominator to be zero. Therefore, the function is continuous on the intervals where \( y eq -1 \). These intervals are \( (-\infty, -1) \) and \( (-1, \infty) \). So, the function is continuous everywhere in these ranges, except at the point where \( y = -1 \).

division by zero

Division by zero is a key reason why certain functions are not continuous at particular points. In rational functions, when the denominator equals zero, the function becomes undefined.

For example, looking at the function \( g(y) = \frac{4}{y+1} \), if \( y = -1 \), then the denominator becomes zero and the function does not have a defined value. In mathematical terms, we say that there is a discontinuity at \( y = -1 \).

It is crucial to check the denominator of rational functions to find where division by zero may occur. By identifying and excluding these points, we ensure that the function is analyzed correctly and understand its intervals of continuity.