Problem 17 - Precalculus with Limits

Question

In Exercises 17-28, evaluate the indicated function for \(f(x) = x^2 + 1\) and \(g(x) = x - 4\). \((f + g)(2)\)

Step-by-Step Solution

Answer

The value of \((f + g)(2)\) is 3.

Step 1: Add the functions

First, we need to perform the operation \(f(x) + g(x)\). This means that we have to add the two given functions: \(x^2 + 1 + (x - 4)\). This simplifies to \(x^2 + x - 3\).

Step 2: Substitute x=2

Next, we need to substitute x = 2 in the result obtained in step 1 to evaluate \((f + g)(2)\). Plugging x=2 into the equation gives us \(2^2 + 2 - 3\).

Step 3: Simplify the equation

Now, we simplify the equation from step 2: \(4 + 2 - 3 = 3\). So, \((f + g)(2) = 3\).

Key Concepts

Function Evaluation

Function evaluation is a method of finding the value of a function for a specific input. Think of it as inserting a number into an equation to see what output you get. For example, with a function defined as \( f(x) = x^2 + 1 \), if you want to find \( f(2) \), replace every \( x \) in the function with \( 2 \).

• Example: \( f(2) = 2^2 + 1 = 4 + 1 = 5 \)
• This shows that when the input is \( 2 \), the output is \( 5 \).

Each function can have its own unique equation or rule. Evaluating functions helps us determine outputs based on specific inputs, assisting us in solving real-world problems.

Polynomial Functions

Polynomial functions are expressions composed of variables raised to non-negative integer powers, summed together with coefficients.\( f(x) = ax^n + bx^{(n-1)} + ... + k \), where \( a, b, ..., k \) are constants. Polynomials can have multiple terms, such as quadratic, linear, and constant terms.

• Quadratic: In the given exercise, \( f(x) = x^2 + 1 \) is a polynomial function, where the highest degree is \( 2 \), making it a quadratic polynomial.
• Linear: Similarly, \( g(x) = x - 4 \) is a linear polynomial with the highest power of \( x \) being \( 1 \).

Understanding these types helps predict their graph shape and behavior. Polynomial functions are versatile tools in modeling various situations.

Combining Functions

Combining functions refers to performing operations like addition, subtraction, multiplication, or division between two functions. It helps create new functions with more complex behaviors. This is what we did in the original exercise with \((f + g)(x)\).

• To combine \( f(x) = x^2 + 1 \) and \( g(x) = x - 4 \), we performed \( f(x) + g(x) \), resulting in \( x^2 + x - 3 \).
• Such operations require you to pay attention to combining like terms.

Through combining, we simplify function expressions to make them easier to work with. This technique is especially useful for analyzing how different elements of multiple functions interact when merged into one.