Problem 11 - Elementary and Intermediate Algebra

Question

Let \(f(x)=\frac{2 x+1}{x^{2}+3 x-4} .\) Find a. \(f(0)\) b. \(f(2)\) c. \(f(1)\)

Step-by-Step Solution

Answer

a. \( f(0) = -\frac{1}{4} \); b. \( f(2) = \frac{5}{6} \); c. \( f(1) \) is undefined.

Step 1: Understanding the Function

The given function is \( f(x) = \frac{2x + 1}{x^2 + 3x - 4} \). It is a rational function, which means it is the division of two polynomials. To find the value of the function at a specific point, we substitute the given value of \( x \) into the function and simplify.

Step 2: Calculating \( f(0) \)

To find \( f(0) \), substitute \( x = 0 \) into the function: \[ f(0) = \frac{2(0) + 1}{0^2 + 3(0) - 4} = \frac{1}{-4}. \] So, \( f(0) = -\frac{1}{4} \).

Step 3: Calculating \( f(2) \)

To find \( f(2) \), substitute \( x = 2 \) into the function: \[ f(2) = \frac{2(2) + 1}{2^2 + 3(2) - 4} = \frac{5}{6}. \] So, \( f(2) = \frac{5}{6} \).

Step 4: Calculating \( f(1) \)

To find \( f(1) \), substitute \( x = 1 \) into the function: \[ f(1) = \frac{2(1) + 1}{1^2 + 3(1) - 4} = \frac{3}{0}. \] Since the function involves division by zero, \( f(1) \) is undefined.

Key Concepts

Polynomials

Polynomials are a fundamental concept in algebra and calculus, and they form the backbone of many mathematical functions, including rational functions. A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents, all combined using addition, subtraction, and multiplication operations. Polynomials can be classified based on the degree—the highest power of the variable present in the expression. For example:

• A linear polynomial has the form: \( ax + b \).
• A quadratic polynomial has the form: \( ax^2 + bx + c \).
• A cubic polynomial has the form: \( ax^3 + bx^2 + cx + d \).

In the given function \( f(x) = \frac{2x + 1}{x^2 + 3x - 4} \), the numerator \( 2x + 1 \) is a linear polynomial, while the denominator \( x^2 + 3x - 4 \) is a quadratic polynomial. Polynomials are essential to constructing rational functions, which are ratios of two polynomials. Understanding polynomials helps in evaluating these functions and analyzing their behavior at different points.

Division by Zero

A division by zero occurs when the denominator of a rational expression equals zero, which is undefined in mathematics. This situation leads to indeterminate forms that cannot be evaluated using standard arithmetic rules. Consider again the function \( f(x) = \frac{2x + 1}{x^2 + 3x - 4} \). To find out where division by zero might occur, set the denominator to zero and solve for \( x \):

• \( x^2 + 3x - 4 = 0 \)
• Factoring gives \((x - 1)(x + 4) = 0\).
• This results in \( x = 1 \) and \( x = -4 \).

The roots \( x = 1 \) and \( x = -4 \) are where the denominator becomes zero, so these are the points where the function \( f(x) \) is undefined. Recognizing these points is crucial in analyzing rational functions, as they illustrate situations where the function cannot produce a valid output. Avoiding division by zero is essential to ensure meaningful evaluations of mathematical expressions.

Evaluating Functions

Evaluating functions involves substituting a specific value in for the variable in a given function and simplifying the expression to find the result. This process is significant for understanding the behavior of functions at particular points. For example, with the function \( f(x) = \frac{2x + 1}{x^2 + 3x - 4} \), consider evaluating \( f(0) \), \( f(2) \), and \( f(1) \):

• For \( f(0) \): Substitute \( x = 0 \) and simplify to get \( f(0) = -\frac{1}{4} \).
• For \( f(2) \): Substitute \( x = 2 \) yielding \( f(2) = \frac{5}{6} \).
• For \( f(1) \): Substitute \( x = 1 \) which results in \( \frac{3}{0} \), an undefined expression due to division by zero.

Each evaluation provides information about the function's behavior at that particular point. They illustrate how the output of a function can vary, depending on whether the conditions lead to defined values or undefined ones, highlighting the importance of careful substitution and simplification when working with functions.