Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \(\left(\frac{f}{g}\right)(5)\)

Step-by-Step Solution

Answer

\((f/g)(5) = 11 / 23\).

Step 1: Compute the result of f(5)

Substitute x = 5 into the function f(x). The equation turns into \(f(5) = 2 * 5 + 1 = 11\)

Step 2: Compute the result of g(5)

Now, substitute x = 5 into the function g(x). The equation becomes \(g(5) = 5^2 - 2 = 23\)

Step 3: Compute the value of (f/g)(5)

Divide the result of function f when x = 5 by the same result for function g. This leads to \((f/g)(5) = f(5) / g(5) = 11 / 23\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It forms the foundation for understanding more complex mathematical concepts and operations.
In the context of the given exercise, algebra is used to express functions, such as \(f(x) = 2x + 1\) and \(g(x) = x^2 - 2\). These functions are algebraic because they involve operations such as multiplication, addition, and exponents with a variable \(x\).
Understanding algebra involves recognizing how to manipulate these symbols and operations to achieve a solution. Here, it requires knowing how to substitute values into the functions, perform calculations, and simplify to find the desired result.
It is essential for students to grasp the basics of algebra to progress into more advanced topics, as it is widely applicable in various fields including science, engineering, and business.

Rational Functions

Rational functions are expressions that represent the ratio of two polynomial functions. In simpler terms, they are fractions with a polynomial numerator and a polynomial denominator.
In the exercise, we deal with the rational function formed by the ratio \(\frac{f}{g}\), which specifically means \(\frac{f(x)}{g(x)}\).
It's crucial to note that rational functions are valid only when the denominator is not zero, to avoid undefined expressions. In our exercise, we must ensure that \(g(x) eq 0\) when evaluating at \(x = 5\), which luckily it isn't, as \(g(5) = 23\).
Understanding rational functions involves knowing how to evaluate each part of the expression, perform the division, and comprehend any restrictions that the denominator might impose.

Substitution Method

The substitution method is a straightforward technique to evaluate functions or solve equations. It involves replacing a variable with a given numerical value to simplify and solve the function or equation.
In the exercise provided, we use substitution to evaluate the individual functions \(f(x)\) and \(g(x)\) at \(x = 5\).

• Start by substituting \(x = 5\) into \(f(x)\), resulting in \(f(5) = 2 \times 5 + 1 = 11\).
• Next, substitute \(x = 5\) into \(g(x)\), resulting in \(g(5) = 5^2 - 2 = 23\).
• Finally, combine these results to compute \((f/g)(5) = \frac{11}{23}\).

By consistently following the substitution method, you simplify the problem into manageable parts. Once values are substituted, arithmetic operations transform functions into simpler numbers that can be further solved or analyzed. This method is incredibly useful in making complex problems approachable and solvable.