Question

Which function assigns the domain value 12 to the range value 7 ? A. \(f(x)=2 x-2\) B. \(f(x)=3 x-4\) C. \(f(x)=\frac{1}{2} x+1\) D. \(f(x)=\frac{3}{7} x+12\)

Short answer

The correct function is C: \( f(x) = \frac{1}{2}x + 1 \)

Step-by-step solution

Understand the Problem

We need to determine which function maps the domain value 12 to the range value 7. In other words, if the input value of the function is 12, the output should be 7.

Substitute x with 12 in each Function

Substitute 12 for x in each of the given functions and solve for f(x).

Check Function A

Substitute 12 into the function A: \( f(x) = 2x - 2 \) \( f(12) = 2(12) - 2 = 24 - 2 = 22 \) Since 22 is not equal to 7, function A is not the correct one.

Check Function B

Substitute 12 into the function B: \( f(x) = 3x - 4 \) \( f(12) = 3(12) - 4 = 36 - 4 = 32 \) Since 32 is not equal to 7, function B is not the correct one.

Check Function C

Substitute 12 into the function C: \( f(x) = \frac{1}{2}x + 1 \) \( f(12) = \frac{1}{2}(12) + 1 = 6 + 1 = 7 \) Since 7 is equal to 7, function C is the correct one.

Verify Function D (Optional)

To be thorough, also check function D: \( f(x) = \frac{3}{7} x + 12 \) \( f(12) = \frac{3}{7}(12) + 12 = \frac{36}{7} + 12 \) \( \frac{36}{7} + 12 = 5.14 + 12 = 17.14 \) Since 17.14 is not equal to 7, function D is not the correct one.

Key concepts

Domain and Range

In mathematics, the domain of a function is the set of all possible inputs (or 'x' values) that the function can accept, while the range is the set of all possible outputs (or 'y' values) the function can produce. Understanding the domain and range is essential for determining the behavior and constraints of a function. For instance, if we have a function like \( f(x) = 2x - 2 \), the domain consists of all real numbers because there's no restriction on 'x'.
The range, however, is also all real numbers because for every possible input, there's a corresponding output. When solving problems, our goal is often to find which function maps a particular domain value to a specific range value.

Substituting Variables

Substituting variables is the process of replacing one variable with a specific value. This is an essential technique for evaluating functions. Let’s break down the process:

• First, identify the function you need to evaluate and the value for the variable.
• Next, replace every instance of that variable within the function with the given value.
• Finally, perform the arithmetic operations according to the function's definition.

For example, if we have the function \( f(x) = 3x - 4 \) and we want to find \( f(12) \), we substitute 12 for every 'x' in the function: \( f(12) = 3(12) - 4 = 36 - 4 = 32 \). This method helps us determine if the given input produces the desired output.

Solving Equations

Solving equations involves finding the value of the variables that satisfy the equation. To determine if a specific function assigns a particular domain value to a specific range value, follow these steps:

• Substitute the given domain value into the function.
• Simplify the function to solve for the range value.
• Compare the calculated range value to the target range value to check if they match.

In our example, we have multiple functions and we need to see which one maps 12 to 7. For \( f(x) = \frac{1}{2}x + 1 \), substituting 12 we get \( \frac{1}{2}(12) + 1 = 7 \), verifying that this function maps 12 to 7. Understanding these steps and practicing substitution will make solving such questions much easier.