Problem 35 - College Algebra

Question

Given the function \(f(x)=8-3 x\) : a. Evaluate \(f(-2)\). b. Solve \(f(x)=-1\).

Step-by-Step Solution

Answer

a. \(f(-2) = 14\); b. \(x = 3\).

Step 1: Substitute x in function

To evaluate \(f(-2)\), substitute \(-2\) for \(x\) in the function equation \(f(x) = 8 - 3x\). This gives us: \(f(-2) = 8 - 3(-2)\).

Step 2: Simplify the expression

Now, simplify the expression \(f(-2) = 8 + 6\). When simplifying, follow the order of operations: multiplication before addition.

Step 3: Calculate the result

Calculate \(f(-2) = 8 + 6\) by performing the addition to find the result \(f(-2) = 14\).

Step 4: Set equation to given value

To solve \(f(x)=-1\), set the function equation equal to \(-1\): \(8 - 3x = -1\).

Step 5: Isolate the variable x

First, move the constant 8 to the other side by subtracting it from both sides: \(-3x = -1 - 8\). Simplify the equation to get \(-3x = -9\).

Step 6: Solve for x

Divide both sides by \(-3\) to solve for \(x\): \(x = \frac{-9}{-3}\). Simplify to find the solution \(x = 3\).

Key Concepts

Function Evaluation

Function evaluation is like inserting a value into a machine to get an output. You have a function, which is a set rule, and you want to see what happens when you input a specific number. In the exercise, you are asked to evaluate \( f(-2) \) with the function \( f(x) = 8 - 3x \). This means you replace every \( x \) in the function with \( -2 \). Thus, it turns into \( f(-2) = 8 - 3(-2) \). Each step in evaluating a function is crucial as it maintains the relationship defined by the function.

Order of Operations

Order of operations is essential when working through expressions to ensure consistent and accurate results. The common acronym used is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the given exercise, after substituting \(-2\) into the expression \(8 - 3(-2)\), you first perform the multiplication part: \(-3 \times -2 = 6\).

• Only after multiplication do you move to addition, turning the expression into \(8 + 6\).

This method helps eliminate mistakes, and when applied correctly, leads to the solution: \( f(-2) = 14 \).

Isolate Variable

Isolating the variable is a common step in solving equations, allowing you to find the unknown value that makes the equation true. To isolate a variable means to get it alone on one side of the equation.

The exercise requires solving \( f(x) = -1 \) for \( x \). We set the function equation \( 8 - 3x = -1 \) and aim to get \( x \) by itself. Start by eliminating constants on the same side as \( x \). Subtract \( 8 \) from both sides to get \( -3x = -9 \). Now, the variable is one step closer to being isolated.

Simplifying Expressions

Simplifying expressions makes equations easier and faster to solve. It involves reducing expressions to their simplest form.

After isolating the variable, you simplify by dividing or canceling terms as necessary to complete solving. In the equation \( -3x = -9 \), divide both sides by \(-3\) to simplify the equation to \( x = 3 \). This simplification uses the operation inverse of multiplication involved in the equation.

• Ensure you've carried out the arithmetic correctly to arrive at this simplest form.

By breaking it down step-by-step, the expression becomes easier to handle, illustrating the power of simplification in solving equations.