Problem 22 - Algebra Form and Function

Question

For what values of \(k\) does the equation \(5-3 x=k\) have a solution? What does your answer say about the range of the function \(f(x)=5-3 x ?\)

Step-by-Step Solution

Answer

Answer: The equation \(5-3x=k\) has a solution for all possible values of \(k\), i.e., its range is the set of all real numbers.

Step 1: Solve the Equation for x

To determine the solution of the given equation \(5-3x=k\), we need to isolate the variable \(x\) by solving for it. To do that, we'll move \(k\) to the left side of the equation and \(3x\) to the right side: \(5-3x-k=0\) Adding \(3x\) to both sides and adding \(k\) to both sides, we get: \(5-k=3x\) Now, divide both sides by \(3\) to isolate \(x\): \(x = \frac{5-k}{3}\)

Step 2: Analyze the Solution's Relation to k

Since we solved for \(x\), we can write the solution as \(x = \frac{5-k}{3}\). Every value of \(k\) corresponds to an \(x\) value where the equation is true. Therefore, the equation \(5-3x=k\) has a unique solution for each possible value of \(k\).

Step 3: Determine the Range of the Function

The range of a function is the complete set of all possible resulting values of the dependent variable, in this case, \(k\). Since there is a unique solution for every possible value of \(k\), there are no restrictions on \(k\). Therefore, the range of \(k\) is all real numbers. The range of the function \(f(x) = 5-3x\) is also the set of all real numbers because the function is a linear function, and linear functions have a continuous range over the set of real numbers.

Step 4: Conclusion

The given equation \(5-3x=k\) has a solution for all possible values of \(k\), i.e., its range is the set of all real numbers. Similarly, the range of the function \(f(x) = 5-3x\) is also the set of all real numbers.

Key Concepts

Range of a Function

The range of a function is the set of all possible output values it can produce. In simpler terms, it's essentially the collection of all the values a function can take as you plug in different inputs. For the linear function \(f(x) = 5 - 3x\), we aim to understand what values \(f(x)\) or \(k\) can take.
Linear functions like \(f(x) = 5 - 3x\) typically do not have restrictions on their range because they represent a straight line on a graph. This straight line extends infinitely in both directions unless bounded by some constraints.
Therefore, since there are no such constraints provided here, \(f(x) = 5 - 3x\) can produce any real number as its output. As a result, the range of this function is all real numbers, denoted as \(\mathbb{R}\).
Key points about function range:

• It's about the outputs (dependent variable).
• Linear functions usually cover all real numbers unless restricted.
• Determining the range helps predict and understand possible scenarios reflected by the function.

Solving for a Variable

Solving for a variable means finding the value of that variable that makes a given equation true. In the equation \(5 - 3x = k\), our goal is to rearrange the equation to isolate \(x\).
First, we move terms around to get all \(x\) terms on one side and constant terms on the other.
We start with:

• Rearranging to \(5 - k = 3x\).
• Then, divide both sides by 3: \(x = \frac{5-k}{3}\).

We see that \(x\) depends on the value of \(k\). This approach shows how solving for a variable involves rearranging and simplifying to get a clear expression for the variable in question. Being able to perform these types of operations is crucial for algebraic problem-solving.

Properties of Linear Functions

Understanding linear functions is essential because they have specific, predictable properties. Let’s look at some important characteristics of linear functions using \(f(x) = 5 - 3x\) as an example.
Main properties include:

• Slope: The number in front of \(x\) (here, \(-3\)) represents the slope, showing the rate at which \(y\) changes concerning \(x\). A negative slope means the line falls as \(x\) increases.
• Y-intercept: The constant term (here, \(5\)) is the y-intercept, signifying where the line crosses the y-axis.
• Continuous Graph: Linear functions graph as straight lines, meaning they are continuous and unbroken.

These properties make it simple to predict and analyze linear functions. They form the foundation for more complex functions used in both mathematics and real-life applications.