Problem 4 - Algebra and Trigonometry

Question

The following questions are about the rational function $$ I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function r has y-intercept ____________

Step-by-Step Solution

Answer

\(\frac{1}{3}\)

Step 1: Identify the y-intercept formula for a rational function

The y-intercept of a function is found by evaluating the function at zero, meaning we set \(x = 0\). For the function \(I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)}\), we substitute \(x = 0\) into the expression.

Step 2: Substitute x = 0 into the rational function

Substitute \(x = 0\) into the function to find its y-intercept: \[ I(0)=\frac{(0+1)(0-2)}{(0+2)(0-3)} \]

Step 3: Simplify the expression

Simplifying the substituted expression: The numerator becomes \((1)(-2) = -2\) and the denominator becomes \((2)(-3) = -6\). So, we have: \[ I(0)=\frac{-2}{-6} = \frac{1}{3} \]

Step 4: Conclusion

Thus, the y-intercept of the function \(I(x)\) is \(\frac{1}{3}\).

Key Concepts

Understanding the Y-Intercept of Rational Functions

The y-intercept of a function is a critical point where the graph intersects the y-axis. For rational functions like \[ I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} \] it's found by evaluating the function at zero, in other words, by setting \( x = 0 \).To find the y-intercept, we substitute zero into the rational function: \[ I(0)=\frac{(0+1)(0-2)}{(0+2)(0-3)} \]This step essentially calculates the function's value when it crosses the y-axis. Remember, at the y-intercept, all rational functions (unless undefined) will present a singular y-value, making it an essential feature of the function's graph.

Evaluating Functions by Substitution

Evaluating a function means finding its value for a specific input. For rational functions, this process involves substituting the given value into the function's equation. Let's see how this works.For example, finding the y-intercept involves evaluating the function at \( x = 0 \). Substituting zero into the function \[ I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} \]turns it into:\[ I(0)=\frac{(0+1)(0-2)}{(0+2)(0-3)} \]Breaking it down, we replace every \( x \) with zero and then simplify the arithmetic within the numerator and the denominator. This technique of substitution is foundational and helps in understanding how the function behaves at particular points. It's a simple but invaluable step in analyzing rational functions.

Simplifying Expressions in Rational Functions

Simplifying expressions is the process of reducing them to their most straightforward form. It's crucial in evaluating functions accurately. When dealing with rational functions, simplification involves both the numerator and the denominator.Let's break down our example:- The numerator: \[ (0+1)(0-2) = 1 \times (-2) = -2 \]- The denominator: \[ (0+2)(0-3) = 2 \times (-3) = -6 \]The next step is to simplify the fraction by dividing the numerator by the denominator:\[ I(0)=\frac{-2}{-6} \]This simplifies to:\[ \frac{1}{3} \]Simplification ensures the expression is in its simplest form, making it easier to interpret and use, especially when determining key features, such as intercepts and asymptotes, of rational functions.