Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the function for \(f(x)=2 x+1\) and \(g(x)=x^{2}-2\) \((f+g)(3)\)

Step-by-Step Solution

Answer

The result of \((f+g)(3)\) is 14.

Step 1: Define the Functions

The function \(f(x)=2x+1\) is a linear function, where 2 is the slope and 1 is the y-intercept. The function \(g(x)=x^{2}-2\) is a quadratic function, with -2 as the y-intercept. These are the functions that we will be using to evaluate \((f+g)(3)\).

Step 2: Substitute 3 into f(x)

First substitute 3 into the function f(x). This would result in \(f(3) = 2(3)+1 = 6+1 = 7\).

Step 3: Substitute 3 into g(x)

Next, substitute 3 into the function g(x). This would result in \(g(3) = (3)^{2}-2 = 9-2=7\).

Step 4: Evaluate (f+g)(3)

Now that we have the results for f(3) and g(3), add these together to find the result of \((f+g)(3)\). This would result in \((f+g)(3) = f(3) + g(3) = 7 + 7 = 14\). So, the answer to \((f+g)(3)\) is 14.

Key Concepts

Understanding Linear Functions

When we talk about linear functions, such as the function presented in the exercise, we are referring to equations that graph a straight line. These functions are written in the form f(x) = mx + b, where m is the slope, and b is the y-intercept. The slope indicates how steep the line is, and the y-intercept is the point where the line crosses the y-axis.

In our example, f(x) = 2x + 1, the slope is 2, which means the line rises two units for every one unit it moves to the right. The y-intercept is 1, indicating that the line crosses the y-axis at the point (0,1). Linear functions are used to model relationships with a constant rate of change. In real-life situations, it could represent things like constant speed or cost per item.

Exploring Quadratic Functions

Quadratic functions introduce a curve into the mix, as they have the form g(x) = ax^2 + bx + c. In these functions, a, b, and c are constants, and x is the variable. The most distinctive feature of a quadratic function is its curved graph, known as a parabola, which can open upwards or downwards depending on the sign of the coefficient a.

For the given function g(x) = x^2 - 2, the coefficient a is 1, which means the parabola opens upwards, and the y-intercept is -2, where the graph crosses the y-axis at (0, -2). Quadratic functions are often used to model situations with acceleration, such as the path of a projectile.

Function Composition and Operation

Function composition involves combining two or more functions in a way that the result of one function is used as the input for another. In our exercise, the operation isn't composition, but rather addition. We're looking at the sum of two functions, expressed as (f+g)(x) = f(x) + g(x). This represents adding the outputs of each function together. To find the sum at a specific point, you evaluate both functions at that point and then add the results.

Performing (f+g)(3) requires calculating both f(3) and g(3), giving us 7 for each based on our linear and quadratic functions, and then adding these values to get 14. This ability to combine functions is a foundational tool in algebra that allows us to explore and solve complex real-world problems.