Problem 39 - Algebra Form and Function

Question

Find the range of \(f\) by finding the values of \(a\) for which \(f(x)=a\) has a solution. $$ f(x)=\frac{5 x+7}{2} $$

Step-by-Step Solution

Answer

Answer: The range of the function is all real numbers.

Step 1: Determine the Function Type

Observe the given function, \(f(x)=\frac{5x+7}{2}\). This is a linear function, which can be rewritten in the form \(y=mx+b\), where \(m\) represents the slope and \(b\) represents the y-intercept. Here m= \(\frac{5}{2}\) and b= \(\frac{7}{2}\).

Step 2: Setting the Function Equal to "a"

The task requires us to find the values of "a" for which \(f(x)=a\) has a solution. Considering \(f(x)=a\), this is equivalent to finding the range of the function, as the range consists of all possible output values. So, let's set the function equal to "a": $$ a = \frac{5x+7}{2} $$

Step 3: Solve for "x" in Terms of "a"

In order to find the range of "a", we will now solve for "x" in terms of "a": $$ a = \frac{5x+7}{2} $$ Multiply both sides of the equation by 2: $$ 2a = 5x+7 $$ Now, isolate x by subtracting 7 from both sides and dividing by 5: $$ x = \frac{2a - 7}{5} $$

Step 4: Analyze the Equation for "x"

Analyzing the equation we derived for "x" in terms of "a": $$ x = \frac{2a - 7}{5} $$ We notice that "a" can take any real number, and there will always be an "x" value that will satisfy the equation. As "a" is not restricted by any conditions, its range is all real numbers.

Step 5: Conclusion

The range of the given function \(f(x)=\frac{5x+7}{2}\) is all real numbers, as "a" can take any real value, and an "x" value will always be found to satisfy the relationship.

Key Concepts

Linear Function

A linear function is a type of equation that creates a straight line when graphed on a coordinate plane. The general form of a linear function is given by \(y = mx + b\), where:

• \(m\) is the slope of the line. This represents the steepness and direction of the line.
• \(b\) is the y-intercept. It indicates the point where the line crosses the y-axis.

In our exercise, the linear function is \(f(x) = \frac{5x + 7}{2}\). We can rewrite this as \(y = \frac{5}{2}x + \frac{7}{2}\). Here, the slope \(m\) is \(\frac{5}{2}\), and the y-intercept \(b\) is \(\frac{7}{2}\).
This means for each unit increase in \(x\), the value of the function increases by \(\frac{5}{2}\) units. Linear functions are straightforward to solve and analyze, making them vital for understanding basic algebraic relationships.

Solving Equations

Solving equations involves finding the value of variables that make the equation true. In this exercise, we are tasked with solving the function equation \(\frac{5x + 7}{2} = a\) to find values of \(x\) in terms of \(a\).
The steps include:

• Multiply both sides by 2 to eliminate the fraction: \(2a = 5x + 7\).
• Isolate \(x\) by subtracting 7 from both sides: \(2a - 7 = 5x\).
• Finally, divide by 5 to solve for \(x\): \(x = \frac{2a - 7}{5}\).

These operations show that as long as \(a\) is a real number, \(x\) will have a corresponding real number value, which helps us define the range. This example highlights the importance of understanding linear equations when trying to find feasible solutions.

Real Numbers

Real numbers are all the numbers on the number line, including both rational and irrational numbers. They encompass:

• Positive numbers, which are greater than zero.
• Negative numbers, which are less than zero.
• Zero itself, which is neutral.

In the context of this exercise, the real numbers play a crucial role in defining the range of the function \(f(x) = \frac{5x + 7}{2}\). Because the solved equation, \(x = \frac{2a - 7}{5}\), can hold for any real number \(a\), it indicates that the outputs (or range) of the function \(f(x)\) are not limited in any way.
Thus, the range includes all real numbers. Understanding this concept clarifies that for any real input, there is a corresponding real output.