Problem 46 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The result of the operation \(3f(x) - 2\) is \(6x + 13\).

Step 1: Identify the correct function

Start by identifying the function that the operation will be performed on. In this case, it is \(f(x)=2x+5\).

Step 2: Perform the multiplication operation

Next, perform the operation on the function. Because the operation is multiplication by 3, the function becomes \(3f(x) = 3(2x+5)\). Simplify this to obtain \(3f(x) = 6x+15\).

Step 3: Subtract 2 from the result

Subtract 2 from the result of step 2. That is, \(3f(x) - 2 = 6x + 15 - 2\). Simplify this to get the final answer, \(3f(x) - 2 = 6x + 13\).

Key Concepts

Understanding Linear Functions

A linear function, like the function \(f(x) = 2x + 5\), can be identified by its linear form \(ax + b\). Here, \(a\) and \(b\) are constants, with \(a\) representing the slope and \(b\) the y-intercept. Linear functions graph as straight lines and exhibit a constant rate of change. The slope \(a\) indicates how steep the line is, and the y-intercept \(b\) tells us where the line crosses the y-axis.
Some key characteristics of linear functions include:

• Constant slope (rate of change)
• Straight-line graph
• Defined for all real numbers

Understanding these components is vital, as it allows us to manipulate and apply these functions in various mathematical contexts.

Exploring Function Operations

Function operations involve performing arithmetic operations such as addition, subtraction, multiplication, and division on functions. In this exercise, we specifically explore multiplication and subtraction involving the function \(f(x)\).
Performing operations on functions means altering the function's equation according to mathematical operations rules. Here, multiplying \(f(x)\) by 3 gives:\[\begin{align*}3f(x) &= 3(2x + 5) \ &= 6x + 15.\end{align*}\]This operation scales the original function, changing both the slope and the intercept proportionately. Following this, subtracting 2 shifts the entire function down, resulting in \(6x + 13\). This demonstrates how function operations can transform both the equation and its graphical representation.

Polynomial Expressions Basics

Polynomial expressions, such as \(g(x)=x^2-3x+2\), consist of terms that are sums of variables raised to non-negative integer powers, each multiplied by a coefficient. They are more complex than linear functions because they can contain one or more terms and higher powers of the variable.
When working with polynomial expressions, it is essential to understand the following:

• Terms are separated by '+' or '-'
• Each term includes a coefficient and a variable raised to an exponent
• The degree of the polynomial is determined by the highest power of the variable

This complexity allows for representations of curves and can include terms like quadratics \((x^2)\), cubes \((x^3)\), and so on. Recognizing these characteristics helps in performing operations and understanding the underlying behavior of polynomial functions.