Question

Use a graphing calculator to determine where \(f(x)=g(x)\). $$f(x)=\frac{2}{3 x}, g(x)=x$$

Short answer

The solutions for \(x\) where \(f(x) = g(x)\) are \(x = \sqrt{\frac{2}{3}}\) and \(x = -\sqrt{\frac{2}{3}}\).

Step-by-step solution

Understanding the functions

It is given that \(f(x)=\frac{2}{3x}\) and \(g(x)=x\). We will first analyze these functions separately. The function \(f(x)\) finds the reciprocal of \(3x\) and multiplies it by 2. The function \(g(x)\) is a line that passes through the origin.

Setting the functions equal

To find the intersection point, we need to set these functions equal to each other, resulting in \(\frac{2}{3x} = x\). This indicates that the x-value for which \(f(x) = g(x)\) is the solution of this equation.

Solving the equation

Solving for \(x\), multiply both sides by \(3x\), giving us \(2 = 3x^2\). Finally, solve for \(x\) by dividing both sides by 3 and taking the square root: \(x = \pm \sqrt{\frac{2}{3}}\).

Checking the solution

Substitute \(x = \sqrt{\frac{2}{3}}\) and \(x = -\sqrt{\frac{2}{3}}\) into \(f(x) = g(x)\), to verify that these solutions make the equation true.

Key concepts

Functions

Functions are mathematical expressions that relate inputs to outputs. In our problem, we have two functions:

• The first function is given by \(f(x) = \frac{2}{3x}\). This function calculates the reciprocal of \(3x\) and then multiplies the result by 2. Essentially, it describes a hyperbola.
• The second function is \(g(x) = x\). This is a simple linear function, which is a straight line through the origin with a slope of 1.

Graphing these functions can help you visually understand how they behave across different values of \(x\). By understanding the shapes and behaviors of these functions, you are better positioned to find where they intersect, which will solve our problem.

Intersection Point

An intersection point is where two functions have the same output value for the same input value \(x\). In other words, it's where their graphs cross each other. For this exercise:

• We need to find the \(x\)-value where \(f(x) = g(x)\).
• Setting \(\frac{2}{3x} = x\) will identify this point by equating both functions.

This critical step involves understanding that the intersection point tells us precisely where these two very different functions, a hyperbola and a line, share a point on the plane.

Solving Equations

To solve for \(x\) when we set \(f(x) = g(x)\), we start with the equation \(\frac{2}{3x} = x\). Here's how to solve it:

• Clear the fraction by multiplying both sides by \(3x\), resulting in the equation \(2 = 3x^2\).
• Next, divide both sides by 3 to isolate \(x^2\): \(x^2 = \frac{2}{3}\).
• Finally, take the square root of both sides to solve for \(x\), which gives \(x = \pm \sqrt{\frac{2}{3}}\).

Sometimes equations can have two solutions, as in this case, where \(x\) could be either positive or negative. Understanding each algebraic manipulation helps in correctly finding these solutions.

Verification of Solutions

After solving for \(x\), it's always crucial to verify that the solutions satisfy the original equation. For our functions:

• Substitute \(x = \sqrt{\frac{2}{3}}\) back into the functions \(f(x)\) and \(g(x)\) to check if they produce the same value.
• Do the same for \(x = -\sqrt{\frac{2}{3}}\).

Both substitutions should hold true, reaffirming these values are actual intersection points of the functions. Verification solidifies our solution and ensures no errors were made during calculations.