Problem 53 - Algebra 2

Question

Find each value if \(f(x)=4 x+3\) and \(g(x)=5 x-7\). $$ g(-0.25) $$

Step-by-Step Solution

Answer

The value is \(-8.25\).

Step 1: Understand the Function

Recognize that you need to find the value of \(g(x)\) when \(x = -0.25\). The function given is \(g(x) = 5x - 7\).

Step 2: Substitute the Value

Substitute \(-0.25\) into the function \(g(x)\). This gives us: \[ g(-0.25) = 5(-0.25) - 7 \]

Step 3: Calculate the Expression inside the Function

First, calculate \(5 \times (-0.25)\). Multiplying gives: \[ 5 \times (-0.25) = -1.25 \]

Step 4: Simplify the Result

Now replace \(5 \times (-0.25)\) with \(-1.25\) in the equation: \[ g(-0.25) = -1.25 - 7 \] Then do the subtraction: \[ -1.25 - 7 = -8.25 \]

Step 5: Conclusion

The value of \(g(-0.25)\) is \(-8.25\).

Key Concepts

Substitution

Substitution is a crucial technique in mathematics used to evaluate functions. It's like replacing a placeholder with a specific value to simplify and solve expressions. In the context of our exercise, we were tasked with finding the value of the function \( g(x) = 5x - 7 \) when \( x = -0.25 \). This means we substitute \( x \) with \( -0.25 \) in the expression.

To perform substitution, follow these steps:

• Identify the variable in the function that needs replacing, which in our case is \( x \).
• Carefully replace each instance of this variable in the function with the given number \( -0.25 \).
• Once replaced, solve the resulting expression step by step to find the desired value.

Substitution might seem simple, but it's the foundation of evaluating and manipulating mathematical expressions.

Linear Functions

Linear functions are a fundamental concept in mathematics, often introduced in early algebra courses. They have the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

In our exercise, the function \( g(x) = 5x - 7 \) is a perfect example of a linear function. **Key characteristics of linear functions include:**

• They graph as straight lines on a coordinate plane.
• The coefficient \( a \) (in this case, 5) represents the slope or steepness of the line.
• The constant \( b \) (here, -7) indicates the Y-intercept, where the line crosses the Y-axis.

Understanding linear functions is important as they represent many real-world relationships where quantities change at constant rates.

Function Notation

Function notation is a way to express the relationship between input values (often \( x \)) and output values (\( f(x) \) or \( g(x) \), etc.). It allows us to represent functions in a concise and formal manner.

**Points to note about function notation:**

• The letter outside the parentheses (for example, \( g \) in \( g(x) \)) is often a way of naming the function.
• The variable inside the parentheses (\( x \) in this case) represents the input or independent variable.
• This notation provides a clear and organized way to substitute different vales into the function, allowing for the calculation of various outputs.

In our exercise, \( g(-0.25) = 5(-0.25) - 7 \) uses function notation to indicate that we are finding the output when \( x = -0.25 \). This notation is beneficial for keeping calculations organized and readable, especially in more complex mathematical problems.