Problem 45 - College Algebra Essentials

Question

Graph \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\log _{\frac{1}{2}} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graph of \(f(x)=\left(\frac{1}{2}\right)^{x}\) is a downward-sloping curve, while \(g(x)=\log _{\frac{1}{2}} x\) goes upwards to the left of \(x = 1\) and downwards to the right. The two functions are inverses of each other and their graphs are reflections of each other across the line \(y=x\).

Step 1: Graph the exponential function

Firstly, graph the function \(f(x)=\left(\frac{1}{2}\right)^{x}\). The base \(\frac{1}{2}\) implies that as \(x\) increases, \(f(x)\) will decrease. Draw a downward-sloping curve starting from high values at negative \(x\) and slowly approaching, but never reaching, the x-axis as we head towards positive \(x\) values.

Step 2: Graph the logarithmic function

Then, graph the function \(g(x)=\log _{\frac{1}{2}} x\). As the base of the logarithm is \(\frac{1}{2}\), the curve will go upwards to the left for positive \(x\) values, then cross the x-axis at \(x = 1\) and continue downwards to the right of \(x = 1\).

Step 3: Observe the symmetry

Lastly, observe the symmetry of the two graphs. They should be mirror images of each other across the line \(y=x\). This confirms that one function is the inverse of the other.