Problem 38 - Algebra 2

Question

Find each value if \(f(x)=3 x-5\) and \(g(x)=x^{2}-x\) \(f\left(\frac{2}{3}\right)\)

Step-by-Step Solution

Answer

The value of \(f\left(\frac{2}{3}\right)\) is \(-3\).

Step 1: Substitute into the Function

We start by substituting the given value into the function. The exercise asks us to find \(f\left(\frac{2}{3}\right)\). Our function is \(f(x) = 3x - 5\). Substitute \(x = \frac{2}{3}\) into the function: \(f\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right) - 5\).

Step 2: Perform Multiplication

Now let's perform the multiplication: compute \(3 \times \frac{2}{3} = 2\). Therefore, the expression simplifies to \(f\left(\frac{2}{3}\right) = 2 - 5\).

Step 3: Finish the Calculation

Lastly, perform the subtraction: \(2 - 5 = -3\). Thus, \(f\left(\frac{2}{3}\right) = -3\).

Key Concepts

Substitution

Substitution is a fundamental concept in algebra where we replace a variable with a specific value or expression given in a problem. It allows us to evaluate functions and expressions by simply plugging in numbers, simplifying, or solving equations. For example, given the function \(f(x) = 3x - 5\), to find \(f\left(\frac{2}{3}\right)\), we substitute \(x\) with \(\frac{2}{3}\). This step transforms our function into \(f\left(\frac{2}{3}\right) = 3 \cdot \frac{2}{3} - 5\).

• Substitution focuses on replacing variables systematically.
• Helps in directly evaluating functions and expressions.
• Is used extensively in algebraic manipulations.

Understanding substitution paves the way for exploring more complex algebraic expressions, helping us see how simple operations can evaluate functions with specific values.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and normal mathematical operations such as addition, subtraction, multiplication, and division. In our example, \(f(x)=3x-5\) is an algebraic expression representing a linear function. To evaluate an algebraic expression, follow these steps:

• Identify the elements of the expression such as coefficients and variables.
• Determine the operations linking these elements.
• Substitute necessary values into the expression.

In the given problem, the algebraic expression \(3x - 5\) helped us evaluate \(f\left(\frac{2}{3}\right)\) by replacing \(x\) with \(\frac{2}{3}\). Recognizing the structure of algebraic expressions allows you to see how parts fit together, creating a foundation for solving equations and evaluating functions efficiently.

Linear Functions

Linear functions are a type of function where the graph is a straight line. These functions typically take the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the problem scenario, you have \(f(x) = 3x - 5\), a linear function that shows a constant rate of change represented by the slope \(m = 3\) and shifts down the y-axis by 5 units, represented by \(b = -5\).

• The slope \(m\) shows how steep the line is.
• The y-intercept \(b\) shows the point where the line crosses the y-axis.
• Linear functions are simple yet powerful tools for modeling relationships between variables.

By understanding the basics of linear functions, you can explore how changes in one variable affect another systematically. Evaluating linear functions involves using substitution and manipulating algebraic expressions, as seen in solving \(f\left(\frac{2}{3}\right)\). This makes them an essential concept in algebra and numerous real-life applications, from calculating costs to predicting trends.