Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f+g)(3)$$

Step-by-Step Solution

Answer

The result of \((f+g)(3)\) is 23.

Step 1: Understand the notation

The notation \((f+g)(x)\) represents the sum of the functions \(f(x)\) and \(g(x)\). This means we need to calculate \((f+g)(x) = f(x) + g(x)\).

Step 2: Find the expressions for f(x) and g(x)

Given the functions, \(f(x) = x^2 + 3x\) and \(g(x) = 2x - 1\). We will add these two expressions together.

Step 3: Add f(x) and g(x)

Compute \((f+g)(x) = f(x) + g(x)\). Substitute the given expressions: \((f+g)(x) = (x^2 + 3x) + (2x - 1)\). Simplify to get \(x^2 + 5x - 1\).

Step 4: Substitute the value x=3 into (f+g)(x)

Now that we have \((f+g)(x) = x^2 + 5x - 1\), we substitute \(x = 3\). So, \((f+g)(3) = 3^2 + 5(3) - 1\).

Step 5: Calculate the result

Evaluate \(3^2 + 5(3) - 1\):- First, compute \(3^2 = 9\).- Next, compute \(5(3) = 15\).- Add these results: \(9 + 15 = 24\).- Finally, subtract 1: \(24 - 1 = 23\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve a sum of powers of a variable, each multiplied by a coefficient. In simple terms, they are numbers with exponents (like squares, cubes) of a variable like \(x\), all added together.
For instance, in the function \(f(x) = x^2 + 3x\), the polynomial consists of two terms:

• \(x^2\), which is a term where the variable \(x\) is raised to the power of 2.
• \(3x\), which is a linear term where \(x\) has an exponent of 1.

Polynomial functions can have one term (monomial), two terms (binomial), or more (polynomial). They are essential in modeling various real-life situations and solving algebraic equations, especially when it comes to graphing curves that represent these functions.
When dealing with polynomial functions, it's important to remember the order of the terms and how they simplify during operations like addition or multiplication.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. They form the building blocks of algebraic mathematics, giving us a way to represent and explore patterns and relationships. For example, in this exercise, we deal with two algebraic expressions:

• \(f(x) = x^2 + 3x\), where \(x^2\) and \(3x\) are terms.
• \(g(x) = 2x - 1\), where \(2x\) is a linear term and \(-1\) is a constant.

Each term in an algebraic expression consists of variables raised to a specific power and multiplied by coefficients, which are numbers that determine the term's line on the graph.
Algebraic expressions can be manipulated through basic operations like addition, subtraction, multiplication, and division, allowing us to solve equations and calculate desired values by plugging in numbers for variables.

Mathematical Operations

Mathematical operations come into play when manipulating expressions, equations, and functions. They include basic actions such as addition, subtraction, multiplication, and division. These operations are crucial in simplifying complex expressions and solving equations.
In the case of function operations, we use methods like addition and composition of functions, as demonstrated in the exercise. Here, we added two functions \(f(x)\) and \(g(x)\) to form a new function \((f+g)(x)\). This process involved combining like terms to simplify the expression:

• Adding \(x^2\) (from \(f(x)\)) and \(2x\).
• Combining \(3x\) and the \(5x\) to get \(x^2 + 5x - 1\).

Once a new expression is formed, substituting specific values (like \(x = 3\)) into it and performing arithmetic operations will give us the function's value at that point.
Understanding these operations is essential in exploring more advanced algebraic concepts and solving complex mathematical challenges.