Problem 43 - Algebra and Trigonometry

Question

Graph \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Step-by-Step Solution

Answer

The graphs of \(f(x)=4^x\) and \(g(x)=log_4 x\) are reflections of each other across the line \(y=x\) due to the property of exponential and logarithmic functions being inverses. The graph of \(f(x)=4^x\) passes through \((0,1)\), \((1,4)\), and \((-1, 0.25)\) while the graph of \(g(x)=log_4 x\) has a vertical asymptote at \(x=0\) and passes through \((1,0)\) and \((4,1)\).

Step 1: Graph the exponential function

To draw \(f(x)=4^x\), remember that exponential functions of the form \(f(x)=a^x\) always pass through the point \((0, 1)\). Since our base is 4, the function will rise rapidly. For \(x=0\), \(y=1\); for \(x=1\), \(y=4\); for \(x=-1\), \(y=0.25\). Plot these values on a graph.

Step 2: Graph the logarithmic function

To draw \(g(x)=log_4 x\), remember that logarithmic functions of the form \(g(x)=log_a x\) have a vertical asymptote at \(x=0\), which implies that the function is undefined for \(x \leq 0\). For \(x=1\), \(y=0\); for \(x=4\), \(y=1\). Plot these points on the same graph where you plotted \(f(x)=4^x\).

Step 3: Identify and explain symmetry

Observe that the two graphs are reflections of each other across the line \(y=x\). This is a consequence of the logarithm and exponential functions being inverses of one another.