Problem 25 - Algebra 1

Question

Evaluate the function when \(x=2, x=0,\) and \(x=-3\). $$ f(x)=0.33 x-2 $$

Step-by-Step Solution

Answer

The values of the function at the given points are \(f(2) = -1.34\), \(f(0) = -2\), and \(f(-3) = -2.99\).

Step 1: Evaluate the function at \(x=2\)

Substitute \(x=2\) into the equation to get \(f(2)=0.33(2)-2\). Then compute the expression to get \(-1.34\).

Step 2: Evaluate the function at \(x=0\)

Substitute \(x=0\) into the equation to get \(f(0)=0.33(0)-2\). Then compute the expression to get \(-2\).

Step 3: Evaluate the function at \(x=-3\)

Substitute \(x=-3\) into the equation to get \(f(-3)=0.33(-3)-2\). Then compute the expression to get \(-2.99\).

Key Concepts

Linear Functions

Understanding linear functions is foundational for algebra and many applications in mathematics. A linear function is a type of function that is represented by a straight line when graphed on a coordinate plane. The standard form of a linear function is given by the equation \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept. The slope \( m \) indicates how steep the line is, and the y-intercept \( b \) indicates the point at which the line crosses the y-axis.

In the context of our exercise, the linear function \( f(x) = 0.33x - 2 \) has a slope of 0.33, meaning that for each unit increase in \( x \) the value of \( f(x) \) increases by 0.33. The y-intercept is -2, indicating that when \( x \) is zero, \( f(x) \) will be -2. This linear function will produce a straight line with a slight upward tilt from left to right and will cross the y-axis at the point \( (0, -2) \).

Function Evaluation

Function evaluation is the process of calculating the output of a function for a particular input value. This involves taking the input value, known as the independent variable (usually \( x \)), and substituting it into the function to find the corresponding output value, known as the dependent variable (usually \( f(x) \)).

To evaluate our given function \( f(x) = 0.33x - 2 \) for a specific value of \( x \) involves replacing the \( x \) in the equation with the value we're interested in. For example, to find the value of the function when \( x = 2 \), we simply plug 2 into the function to get \( f(2) = 0.33(2) - 2 \), and then calculate the result. The evaluation process looks the same for any value of \( x \), and allows us to graph the function or find specific points of interest on the function's graph.

Substitution Method

The substitution method is a technique used in algebra to solve equations or evaluate functions by replacing variables with their corresponding values. When a function such as \( f(x) \) is given, and you're asked to find the value for certain \( x \) values, the substitution method is the primary tool used.

Let's look at the steps involved in our exercise using the substitution method:

1. Identify the value to substitute for \( x \).
2. Replace \( x \) in the function \( f(x) \) with the identified value.
3. Simplify the function by performing the necessary arithmetic operations.
4. Write down the resulting value, which is \( f(x) \) for the given \( x \) value.

For instance, to find \( f(-3) \) using the substitution method, we replace \( x \) in the function with -3, giving us: \( f(-3) = 0.33(-3) - 2 \). We then simplify it to obtain \( -2.99 \), which is the function's output when \( x = -3 \). This straightforward process is repeated for various \( x \) values to understand the function's behavior or to solve for unknowns.