Problem 37 - Algebra 2

Question

Find each value if \(f(x)=3 x-5\) and \(g(x)=x^{2}-x\) \(g\left(\frac{1}{3}\right)\)

Step-by-Step Solution

Answer

\(g\left(\frac{1}{3}\right) = \frac{-2}{9}\).

Step 1: Identify the Function

We need to find the value of the function \(g(x)\) at \(x = \frac{1}{3}\). The function given is \(g(x) = x^2 - x\).

Step 2: Substitute the Value

Substitute \(x = \frac{1}{3}\) into the function \(g(x)\). This gives us: \[ g\left(\frac{1}{3}\right) = \left(\frac{1}{3}\right)^2 - \left(\frac{1}{3}\right) \]

Step 3: Calculate \(x^2\) for \(x = \frac{1}{3}\)

Calculate \(\left(\frac{1}{3}\right)^2\): \[ \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

Step 4: Evaluate \(-x\) for \(x = \frac{1}{3}\)

Evaluate \(-\left(\frac{1}{3}\right)\): \[ -\left(\frac{1}{3}\right) = -\frac{1}{3} \]

Step 5: Simplify and Find \(g\left(\frac{1}{3}\right)\)

Substitute the evaluated parts back into the equation from Step 2: \[ g\left(\frac{1}{3}\right) = \frac{1}{9} - \frac{1}{3} \]Convert \(-\frac{1}{3}\) to a fraction with denominator \(9\): \( -\frac{1}{3} = -\frac{3}{9} \).Subtract \(\frac{3}{9}\) from \(\frac{1}{9}\) to find the solution: \[ g\left(\frac{1}{3}\right) = \frac{1}{9} - \frac{3}{9} = \frac{-2}{9} \]

Key Concepts

Understanding Algebraic Functions

Algebraic functions are expressions that involve algebraic operations like addition, subtraction, multiplication, and division, as well as roots and powers. They are fundamental in algebra, serving as building blocks for more complex equations. In our exercise, we work with two algebraic functions:

• The first function, \(f(x) = 3x - 5\), is a linear function, meaning it graphs as a straight line, with a slope of 3 and a y-intercept at -5.
• The second function, \(g(x) = x^2 - x\), is a quadratic function, recognized by the highest power, \(x^2\). It graphs as a parabola, where the shape of the curve opens upwards, forming a U-like structure.

Both functions take an input value, \(x\), and perform a series of operations to output a result. Understanding how each operation affects the graph is critical.
Algebraic functions allow us to explore relationships between variables, analyze graphs, and solve real-world problems through mathematical modeling.

How to Use the Substitution Method Efficiently

The substitution method is a key process for evaluating functions at specific input values. This involves replacing the variable in a function with a particular number, allowing us to find the corresponding function value.
In our exercise, we're tasked with finding the value of the function \(g(x) = x^2 - x\) for \(x = \frac{1}{3}\). Here's a deeper look at how it's done:

• First, identify the value you need to substitute. In this case, it's \(x = \frac{1}{3}\).
• Replace every instance of \(x\) in the function \(g(x)\) with \( \frac{1}{3} \). This substitution creates a new expression that's purely numerical: \[ g\left(\frac{1}{3}\right) = \left(\frac{1}{3}\right)^2 - \frac{1}{3} \]

By using the substitution method, you simplify the function into a format that can be solved with basic arithmetic, making it a powerful tool for solving and analyzing algebraic functions.

Steps to Simplify Expressions Effectively

Simplifying expressions is about making complex expressions easier to understand and solve. It's a crucial part of algebra, as it enables clearer insights and solutions to math problems.
When simplifying algebraic expressions, here are some key steps to keep in mind:

• Combine like terms: Terms in an expression that have the same variables raised to the same power can be combined. For example, \(3x + 2x = 5x\).
• Convert fractions: To subtract fractions like \(\frac{1}{9} - \frac{1}{3}\), convert them to have a common denominator. In our exercise, we convert \(\frac{1}{3}\) to \(\frac{3}{9}\), enabling the subtraction.
• Perform operations systematically: Calculate powers and perform additions/subtractions carefully to avoid mistakes. This includes solving \(\left(\frac{1}{3}\right)^2\) as \(\frac{1}{9}\) clearly before proceeding with the next steps.

Ultimately, simplifying expressions makes it possible to understand and solve algebraic problems more easily and accurately, leading to efficient solutions.