Question

Sketch the graph of the function. $$ f(x)=3^{|x|} $$

Short answer

The correct sketch of the function \(f(x) = 3^{|x|}\) should be a reflection of the graph of \(y = 3^x\) over the y-axis, forming a shape like a 'V'. Starting from the point (0,1), it increases towards infinity as it moves away in either positive or negative x-direction.

Step-by-step solution

Understanding the Function

The function is \(f(x) = 3^{|x|}\). This is an exponential function, where the variable x is under an absolute value operation. This absolute value operation means that the function will always take the positive value of x.

Plot the graph for positive x

Since \(f(x) = 3^x\) for \(x > 0\), graph this function as you would any other exponential function, starting at (0,1) and increasing towards infinity as x increases.

Reflect the graph for negative x

For negative x, the absolute value function gets rid of the negative sign, turning it into positive. This means that we just reflect the graph over the y-axis. This reflection will yield the graph for \(x < 0\). Since we redefined negative x values as positive, the exponential function behaves the same way as it does for positive x, and outputs the same y values.

Combine the Graphs

Now combine the two graphs from step 2 and step 3. This creates a graph that starts at (0,1), increases towards infinity as x moves away in either the positive or negative x-direction.