Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The function \(f(x)=4^{x}\) increases steeply for positive \(x\) and approaches the \(x\)-axis as a horizontal asymptote for negative \(x\). On the contrary, the function \(g(x)=\log _{4} x\) has the \(y\)-axis as a vertical asymptote and increases as \(x\) becomes large. Both the functions intersect at points (1, 1) and (4, 2) and are reflections of each other in the line \(y=x\), indicative of their inverse properties.

Step 1: Graphing \(f(x)=4^{x}\)

To graph the function \(f(x)=4^{x}\), note that it's an exponential function. \n\nAt \(x = 0\), \(f(x) = 4^0 = 1\). This is a point on the graph. As \(x\) approaches infinity (\(x → ∞\)), \(f(x)\) also approaches infinity as anything to the power infinity yields infinity. Similarly, As \(x\) approaches negative infinity (\(x → -∞\)), the function value approaches zero (\( f(x) → 0\)). Therefore, the graph will increase steeply for positive \(x\) and approach the \(x\)-axis as a horizontal asymptote for negative \(x\). Plot the function accordingly.

Step 2: Graphing \(g(x)=\log _{4} x\)

To graph \(g(x)=\log _{4} x\), note that it's a logarithmic function with base 4. \n\nWhen \(x = 1\), \( g(x) =\log _{4}(1) = 0\), and when \(x = 4\), \( g(x) =\log _{4}(4) = 1\). These are two points on the graph. As \(x\) approaches infinity (\(x → ∞\)), \(g(x)\) also approaches infinity (\(g(x) → ∞\)), and as \(x\) approaches 0 (\(x → 0\)), \(g(x)\) approaches negative infinity (\(g(x) → -∞\)). This indicates that the \(y\)-axis is a vertical asymptote. Plot the function accordingly.

Step 3: Combining the Graphs

Add both graphs to the same coordinate system. The graph of \(f(x)=4^{x}\) should intersect with the graph of \(g(x)=\log _{4} x\) at the point (1,1) and (4,2) as they are the points where both functions are equal. They should also reflect about the line \(y=x\) as they are inverse of each other.