Problem 4 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The y-intercept is \( \frac{1}{3} \).

Step 1: Find the y-intercept Formula

The y-intercept of a function is the y-value of the function when x is zero. For the function \( r(x) \), this means evaluating \( r(0) \).

Step 2: Substitute x=0 into the Function

Substitute \( x = 0 \) into the function \( r(x) = \frac{(x+1)(x-2)}{(x+2)(x-3)} \). This results in evaluating \( r(0) = \frac{(0+1)(0-2)}{(0+2)(0-3)} \).

Step 3: Simplify the Expression

Simplify the expression:- The numerator becomes \((1)(-2) = -2\).- The denominator becomes \((2)(-3) = -6\). Thus, \( r(0) = \frac{-2}{-6} \).

Step 4: Simplify the Fraction

Simplify the fraction \( \frac{-2}{-6} \) by dividing the numerator and the denominator by their greatest common divisor, which is 2. This simplifies to \( \frac{1}{3} \).

Step 5: Write the Final Answer

The y-intercept of the function \( r(x) = \frac{(x+1)(x-2)}{(x+2)(x-3)} \) is \( \frac{1}{3} \).

Key Concepts

y-intercept

The y-intercept of a function is found where the graph crosses the y-axis. This happens when the value of x is zero. To find the y-intercept of a function like a rational one, we substitute x = 0 into the function and solve for y. In this instance, for the function \( r(x) = \frac{(x+1)(x-2)}{(x+2)(x-3)} \), substituting 0 for x, we get \( r(0) = \frac{(0+1)(0-2)}{(0+2)(0-3)} \). This substitution simplifies the function and lets us evaluate it at a specific point without needing complicated graph analysis.

function evaluation

Evaluating a function involves substituting a specific value for each variable in the function. For a rational function like \( r(x) = \frac{(x+1)(x-2)}{(x+2)(x-3)} \), evaluating r at x = 0 means replacing every instance of x with 0. Function evaluation allows us to find specific points on the graph of the function. In this case, it helps us to find the y-intercept by calculating \( r(0) = \frac{1 \cdot (-2)}{2 \cdot (-3)} \), which is \( \frac{-2}{-6} \). By plugging in the numbers, you get a numerical result that makes the problem-solving process straightforward.

simplifying expressions

Simplifying an expression means reducing it to its simplest form. This often involves combining like terms or reducing fractions. In our example, after substituting x = 0, we have the expression \( \frac{(-2)}{(-6)} \). Each term is multiplied or divided as needed to make the expression more manageable. In the expression \( 1 \cdot (-2) = -2 \) and \( 2 \cdot (-3) = -6 \), simplifying means ensuring that both the numerator and denominator are reduced as much as possible, which prepares the expression for further simplification.

fractions

Fractions combine a numerator and a denominator to represent a part of a whole. Simplifying a fraction like \( \frac{-2}{-6} \) involves identifying the greatest common divisor (GCD) of both numbers. In this case, the GCD is 2. To simplify the fraction, you divide both the top and bottom by this number: \( \frac{-2}{-6} = \frac{1}{3} \). The result is a fraction that represents the same quantity but in the simplest form, making it easier to understand and work with further. Learning to simplify fractions is a valuable skill that is frequently used in math.