Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). Given the function \(f(x)=8-3 x:\) (a) Evaluate \(f(-2)\) (b) Solve \(f(x)=-1\).

Step-by-Step Solution

Answer

\(f(-2) = 14\); solution to \(f(x) = -1\) is \(x = 3\).

Step 1: Evaluate f(-2)

Substitute \(-2\) into the function \(f(x) = 8 - 3x\) to find \(f(-2)\).\\\[\ f(-2) = 8 - 3(-2)\]\\Simplify the expression: \[\ f(-2) = 8 + 6\]\\[\ f(-2) = 14\]\.\Thus, the value of \(f(-2)\) is \ 14\.

Step 2: Solve for f(x) = -1

Set the function equal to \(-1\) and solve for \(x\). Replace \(f(x)\) with \(-1\).\\\[\ 8 - 3x = -1\]\\Add \(3x\) to both sides: \[\ 8 = 3x - 1\]\\Add \(1\) to both sides: \[\ 8 + 1 = 3x\]\\[\ 9 = 3x\]\\Divide both sides by \(3\) to solve for \(x\): \[\ x = 3\]\.Thus, \(x = 3\) is the solution to \(f(x) = -1\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They form the backbone of algebra and are used to define mathematical relationships. In the function provided, \( f(x) = 8 - 3x \), you can see that it is an algebraic expression composed of the constant 8 and the term \(-3x\), where 8 is a constant, and \(-3x\) represents a linear term with a coefficient of \(-3\). The variable \(x\) is the unknown that we manipulate in various ways to solve problems.Understanding algebraic expressions involves recognizing the parts of the expression and how they interact. In this case, the expression shows a linear relationship where changing \(x\) affects the output value of the function, \(f(x)\). Algebraic expressions can represent anything from simple arithmetic problems to complex real-world scenarios. By substituting different values of \(x\) into the expression, we can evaluate different scenarios and results related to the function.

Solving Equations

Solving equations is the process of finding the values of variables that satisfy an equation. In the given problem, we needed to solve the equation \(f(x) = -1\). The original function was \(f(x) = 8 - 3x\). By setting it equal to \(-1\), we seek to find the value of \(x\) that makes the equation true.Here are the steps to solve the equation:

• Set the equation: \(8 - 3x = -1\).
• Add \(3x\) to both sides to isolate the constant on one side: \(8 = 3x - 1\).
• Next, add 1 to both sides to further simplify: \(8 + 1 = 3x\).
• Now, the equation becomes \(9 = 3x\).
• Finally, divide both sides by 3 to solve for \(x\): \(x = 3\).

Through this process, we found that \(x = 3\) is the solution. Solving equations helps us find unknown values by manipulating the terms to isolate the variable.

Substitution Method

The substitution method involves replacing variables with given numbers or expressions to evaluate or simplify an expression or equation. It is especially useful in evaluating functions like in this exercise where you calculate values such as \(f(-2)\).To evaluate \(f(-2)\), follow these steps:

• First, substitute \(-2\) for \(x\) in the expression: \(8 - 3(-2)\).
• Simplify inside the parentheses: \(8 - (-6)\).
• This simplifies further to \(8 + 6\).
• Calculate the final result: \(14\).

By substituting \(-2\) into the function \(f(x) = 8 - 3x\), we determine that \(f(-2) = 14\). Using substitution, we can quickly evaluate expressions and functions by replacing variables with actual values.