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Chapter 12: Problem 43

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

Short Answer

Expert verified
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 by step solution

01

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 鈫� 鈭瀄)), \(f(x)\) also approaches infinity as anything to the power infinity yields infinity. Similarly, As \(x\) approaches negative infinity (\(x 鈫� -鈭瀄)), the function value approaches zero (\( f(x) 鈫� 0\)). 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.
02

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 鈫� 鈭瀄)), \(g(x)\) also approaches infinity (\(g(x) 鈫� 鈭瀄)), and as \(x\) approaches 0 (\(x 鈫� 0\)), \(g(x)\) approaches negative infinity (\(g(x) 鈫� -鈭瀄)). This indicates that the \(y\)-axis is a vertical asymptote. Plot the function accordingly.
03

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.

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