Question

You can obtain a graphical representation of the relationship \(2^{1 / 2}=\sqrt{2}\) by investigating the graph of \(f(x)=2^{x}\) a. Graph \(f(x)=2^{x}\) b. Use the Trace feature to find values of \(f\) when \(x=\frac{1}{2}\) c. Compare the value from part (b) with the value of \(\sqrt{2}\).

Short answer

On graphing \(f(x)=2^{x}\) and tracing \(x=\frac{1}{2}\), we get \(f(\frac{1}{2})\) approximately equal to 1.414. This value is the same as \(\sqrt{2}\), showing that \(2^{\frac{1}{2}}\) is indeed equal to \(\sqrt{2}\).

Step-by-step solution

Graph the Function f(x)=2^x

Start by graphing the function \(f(x)=2^{x}\). You can use a graphing tool or calculator for this. The graph should show that the function is increasing and that it crosses the y-axis at 1.

Use the Trace Feature for x=1/2

Next, use the Trace feature to find values of \(f\) when \(x=\frac{1}{2}\). The Trace feature allows you to move along the curve of the graph and obtain the y-coordinate corresponding to the chosen x-coordinate. When you trace to \(x=\frac{1}{2}\), note the corresponding value of \(f(x)\).

Compare the Value with √2

Finally, compare the obtained value from Step 2 with \(\sqrt{2}\). The square root of 2 is approximately 1.414, and you'll observe that \(f(x)\) at \(x=\frac{1}{2}\) also has the same numerical value.