Problem 2 - College Algebra

Question

If \(f(x)=3 x^{2}-2 x+1\) and \(g(x)=4 x-1,\) find: a. \((f+g)(x)\) b. \((f+g)(5)\)

Step-by-Step Solution

Answer

a) The sum of the functions, (f+g)(x), is 3x^2 + 2x. b) The value of this combined function when evaluated at x=5 is 85.

Step 1: Find the sum of the functions

The sum of two functions, (f+g)(x), is the sum of the respective functions. Therefore, add the two functions together to get \( (f+g)(x) = f(x) + g(x) = (3 x^2 - 2 x + 1) + (4 x - 1)\). Simplifying, this results in \( (f+g)(x) = 3x^2 + 2x \).

Step 2: Evaluate the sum of the functions at x=5

Substitute x=5 into the combined function found in step 1, \( (f+g)(5) = 3(5)^{2} + 2(5) \). Simplifying, this results in \( (f+g)(5) = 75 + 10 = 85 \).

Key Concepts

Polynomial Functions

Polynomial functions are expressions composed of variables and coefficients, connected by addition, subtraction, and multiplication, but never division by a variable. These functions are written in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each term has a non-negative integer exponent.

• The highest exponent of the variable is called the degree of the polynomial.
• In our exercise, \( f(x)=3x^2-2x+1 \) is a quadratic polynomial because the highest exponent of \(x\) is 2.

Polynomial functions can change their appearance based on operations such as addition or subtraction. These characteristics help in modeling various real-world scenarios where relationships can be expressed through equations.

Function Evaluation

Function evaluation refers to the process of finding the value of a function for a specific input. Function evaluation is crucial in mathematics and its applications because it helps in understanding the behavior of functions.
When evaluating a function, follow these steps:

• Identify the function expression, for instance, \((f+g)(x) = 3x^2 + 2x\).
• Substitute the given input value into the function. In our case, when \(x = 5\), substitute to get \((f+g)(5) = 3(5)^2 + 2(5)\).
• Simplify the expression to find the numerical value. This results in \((f+g)(5) = 75 + 10 = 85\).

Function evaluation is a powerful tool for verifying results and exploring mathematical models.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (addition, subtraction, multiplication, and division) that represent a particular value or range of values.
Key elements of algebraic expressions include:

• Variables: symbols like \(x\) or \(y\) that represent unknown or changeable values.
• Constants: fixed numbers such as \(3\) or \(-1\) in the expressions.
• Coefficients: numbers that multiply the variables, like \(3\) in \(3x^2\).
• Operators: such as \(+\), \(-\), and \(\times\), which connect the terms together.

In algebra, understanding how to manipulate these elements enables solving equations and exploring mathematical concepts. The exercise we explored demonstrates how to combine and simplify algebraic expressions through function operations.