Problem 9 - Algebra 2

Question

Let \(f(x)=3 x+5\) and \(g(x)=x^{2} .\) Perform each function operation. $$ (g-f)(x) $$

Step-by-Step Solution

Answer

The result is \( (g-f)(x) = x^2 - 3x - 5\)

Step 1: Define the functions

First, define the two functions. We have \(f(x) = 3x + 5\) and \(g(x) = x^2\)

Step 2: Compute (g-f)(x)

Then we can compute \(g(x) - f(x)\) by subtracting the entire function f(x) from g(x). That means, for every value of x, the value of function g(x) will be reduced by the value of function f(x). So, the new function (g-f)(x) will be equal to \(g(x) - f(x) = x^2 - (3x + 5)\).

Step 3: Simplify the result

Finally, we can simplify the expression by removing the brackets. This will result in \(x^2 - 3x - 5\)

Key Concepts

Understanding the Difference of Functions

The difference of functions is a common operation where you take two functions and find the result of subtracting one from the other. In mathematical terms, if you have two functions, let's call them \(f(x)\) and \(g(x)\), you can find their difference by calculating \((g-f)(x) = g(x) - f(x)\). This operation involves subtracting the values of \(f(x)\) from \(g(x)\) for each possible input value \(x\). This is useful for comparing how two functions behave relative to each other across their domain.

When computing the difference:

• Ensure all operations are properly applied, preserving signs throughout.
• Simplify the resulting expression by combining like terms.

In our exercise, we started with \(g(x) = x^2\) and \(f(x) = 3x + 5\). The difference, \((g-f)(x)\), translates to \(x^2 - (3x + 5)\). Simplifying this expression involves distributing the negative sign to both \(3x\) and \(5\), resulting in \(x^2 - 3x - 5\). This gives us a clear picture of how the differences between \(g(x)\) and \(f(x)\) vary with different \(x\) values.

Grasping Function Notation

Function notation is a convenient way to express the idea that something is a function of one variable, usually \(x\). This notation is typically written as \(f(x)\), where \(f\) is the name of the function and \(x\) is the variable. The expression inside the parentheses (\(x\)) shows what the variable is, and it can change which in turn changes the output of the function.

Function notation helps us:

• Clearly communicate which variable influences the function's output.
• Visually distinguish between different functions like \(f(x)\), \(g(x)\), and \(h(x)\).
• Manipulate functions easily, especially when taking differences, sums, or compositions.

In our exercise, function notation let us clearly define \(f(x) = 3x + 5\) and \(g(x) = x^2\). This notation made it straightforward to calculate and express the difference \((g-f)(x)\) by using a simple, recognizable format. As you work with function operations, being comfortable with this notation will help you understand and solve problems more effectively.

Basics of Polynomial Functions

Polynomial functions are a key concept in algebra, comprising expressions that involve sums of powers of variables multiplied by coefficients. They generally have the form \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(n\) is a non-negative integer, and \(a_n, a_{n-1}, ..., a_1, a_0\) are constants.

Some characteristics of polynomial functions include:

• They are continuous and smooth curves, which makes them easy to plot and analyze on a graph.
• The degree of the polynomial (highest power of \(x\)) determines its behavior as \(x\) tends to infinity or negative infinity.
• They can be added, subtracted, multiplied, and even divided (quotients of polynomials are called rational functions).

In our example, both \(g(x) = x^2\) and \(f(x) = 3x + 5\) are polynomial functions. \(g(x)\) is a quadratic polynomial because its highest degree is 2, and \(f(x)\) is a linear polynomial as its highest degree is 1. When we found \(g(x) - f(x) = x^2 - 3x - 5\), the result is a polynomial of degree 2, echoing the fact that polynomials behave predictably under basic operations like addition and subtraction.