Problem 22 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The value of (f+g)(1) is 2

Step 1: Evaluate function f(1)

Substitute x=1 into function f(x), which results in \(f(1)=2(1) + 1 = 3\)

Step 2: Evaluate function g(1)

Next, substitute x=1 into function g(x), which results in \(g(1)=(1)^2 - 2 = -1\)

Step 3: Evaluate (f+g)(1)

Now, use the results from Step 1 and Step 2 to compute (f+g)(1). This is given by \(f(1) + g(1) = 3 + (-1) = 2\)

Key Concepts

Function Evaluation

Function evaluation is a fundamental aspect of understanding functions in mathematics. It involves finding the output of a function given an input value. Let's consider the example provided with the functions f(x) and g(x). To evaluate f(1), we simply substitute x with 1 in the expression for f(x), resulting in f(1) = 2(1) + 1, which simplifies to 3. Similarly, to find g(1), we replace x with 1 in g(x), yielding g(1) = (1)^2 - 2 which simplifies to -1.

Understanding function evaluation is crucial for working with function compositions, as the output of one function often becomes the input for another. When you master this concept, you can solve complex problems by breaking them down into simpler, more manageable parts.

Polynomial Functions

Polynomial functions comprise a category of functions that are particularly versatile in various fields of mathematics and science. They have the general form p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where n is a non-negative integer and the a_i's are real numbers. In our example, the function g(x) = x^2 - 2 is a polynomial, specifically a quadratic function, which is a polynomial of degree 2.

Understanding polynomials is significant because they can model many types of real-world phenomena. From physics to economics, polynomial functions are used to describe parabolic trajectories, trends, and much more. Recognizing the shape and properties of these functions allows for predictions and calculations that are essential in various applications.

Arithmetic Operations

Arithmetic operations are the cornerstone of basic mathematics. They include operations such as addition, subtraction, multiplication, and division. In the context of functions, arithmetic operations can be applied to combine or modify functions. In the exercise, we add the outputs of the two functions f(x) and g(x) to find (f+g)(x).

By performing the arithmetic operation of addition, we can find the combined effect of both f(1) and g(1). This process demonstrates not only the art of combing functions but also the real essence of forming new functions from existing ones. Arithmetic operations on functions enable us to construct complex models and solutions from simpler, more fundamental pieces.