Problem 21 - Algebra 1

Question

Evaluate the function when \(x=2, x=0,\) and \(x=-3\). $$ g(x)=8 x-2 $$

Step-by-Step Solution

Answer

The function evaluates to 14, -2, and -26 at x=2, x=0, and x=-3, respectively.

Step 1: Substitute \(x=2\)

The function \[g(x)=8x-2\] can be evaluated for \(x=2\) by substituting \(x\) with 2. Doing so results in \[g(2)=(8*2)-2=14\]

Step 2: Substitute \(x=0\)

The function \[g(x)=8x-2\] can be evaluated for \(x=0\) by substituting \(x\) with 0. Doing so results in \[g(0)=(8*0)-2=-2\]

Step 3: Substitute \(x=-3\)

The function \[g(x)=8x-2\] can be evaluated for \(x=-3\) by substituting \(x\) with -3. Doing so results in \[g(-3)=(8*-3)-2=-26\]

Key Concepts

Substitution in Functions

Substitution is one of the basic techniques used in mathematics to evaluate functions. It involves replacing a variable with a numerical value to find the output of a function. In the context of the given exercise, the function is defined as \( g(x) = 8x - 2 \). When you're asked to evaluate this function for different values of \( x \), you simply substitute each value into the function.

• For \( x = 2 \), substitute 2 into \( g(x) \) to get \( g(2) = 8(2) - 2 \).
• For \( x = 0 \), substitute 0 into \( g(x) \) to get \( g(0) = 8(0) - 2 \).
• For \( x = -3 \), substitute -3 into \( g(x) \) to get \( g(-3) = 8(-3) - 2 \).

Each substitution changes the variable \( x \) into a specific number, allowing you to simplify the expression to find the result. This process is crucial for evaluating functions and solving problems in algebra.

Understanding Linear Functions

Linear functions are a type of function where the graph of the solutions is a straight line. They can be expressed in the standard form \( f(x) = mx + b \), where \( m \) and \( b \) are constants. In the exercise, the function \( g(x) = 8x - 2 \) is a linear function.

• \( m \) (the coefficient of \( x \)) is 8. This represents the slope, showing the rate of change of the function. For every unit increase in \( x \), \( g(x) \) increases by 8 units.
• \( b \) is the constant term, -2, which is the \( y \)-intercept. This is where the line crosses the y-axis.

These components make linear functions easy to work with. They follow a predictable pattern, allowing for quick calculations and understanding of their behavior. Evaluating a linear function simply involves using substitution to find corresponding values of \( g(x) \) for different \( x \) inputs.

Demystifying Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the function \( g(x) = 8x - 2 \), the algebraic expression involves multiplication, subtraction, and variables.

• "8x" is the term where the variable \( x \) is multiplied by the constant 8. This demonstrates how variables are used to represent numbers that can change.
• The "-2" is a constant term, showing how arithmetic operations are combined with variables to form expressions.

The goal of working with algebraic expressions is to simplify them or solve them by finding the values of the variables that make the equation true. Substitution helps to achieve this by directly inputting known values for the variables, turning the expression into a simple arithmetic problem which can be easily evaluated.