Problem 51 - Algebra 2

Question

Find each value if \(f(x)=4 x+3\) and \(g(x)=5 x-7\). $$ g(-1) $$

Step-by-Step Solution

Answer

The value of \( g(-1) \) is \(-12\).

Step 1: Understanding the Problem

To solve for the value of \( g(-1) \), we need to determine the output of the function \( g(x) \) when \( x = -1 \). Use the function formula \( g(x) = 5x - 7 \).

Step 2: Substitution

Substitute \( x = -1 \) into the function \( g(x) = 5x - 7 \). This yields \( g(-1) = 5(-1) - 7 \).

Step 3: Simplifying the Expression

Calculate \( 5(-1) \) which equals \(-5\). Then subtract 7 from \(-5\), giving \( -5 - 7 \).

Step 4: Calculate the Final Result

Solve \( -5 - 7 = -12 \). Thus, \( g(-1) = -12 \).

Key Concepts

Substitution

Substitution is a key mathematical technique used frequently when working with functions and equations. It involves replacing variables with specific values in order to evaluate or simplify expressions. In this exercise, we used substitution to find the value of the function \( g(x) \) when \( x = -1 \).

Here's how the substitution worked in our problem:

• We identified the value to substitute, which is \( x = -1 \).
• We then inserted this value into the given function formula for \( g(x) \), yielding \( g(-1) = 5(-1) - 7 \).
• This substitution allows us to simplify and solve for the function output.

Substitution is an essential step in solving functions because it directly connects the abstract function definition to specific, tangible outputs.

Linear Function

A linear function is a type of function that graphically represents a straight line. It typically takes the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. In this exercise, we dealt with two linear functions: \( f(x) = 4x + 3 \) and \( g(x) = 5x - 7 \).

Understanding the characteristics of a linear function is crucial:

• The \( a \) value, or coefficient of \( x \), determines the slope of the line. For instance, \( g(x) \) has a slope of 5, indicating it rises steeply.
• The \( b \) value, referred to as the y-intercept, shows where the line crosses the y-axis. In \( g(x) \), this is at -7.

Linear functions are widely used in various fields because of their simplicity and effectiveness in modeling constant rates of change.

Function Evaluation

Function evaluation is the process of determining the output of a function for a particular input value. This process is essential for understanding what a function describes within a given context. In our example, evaluating the function \( g(x) = 5x - 7 \) at \( x = -1 \) helps us find the specific output associated with that input.

The steps to evaluate a function are straightforward:

• First, identify the input value, in this case, \( x = -1 \).
• Next, substitute this value into the function to replace \( x \), which results in \( g(-1) = 5(-1) - 7 \).
• Finally, perform the arithmetic operations to simplify and find the output, which was -12 in our case.

Function evaluation is a fundamental aspect of working with functions, allowing for precise answers that are crucial in mathematical problem-solving.