Problem 75 - Precalculus

Question

$$\text { Graph } f(x)=e^{|x|}$$

Step-by-Step Solution

Answer

The graph is symmetric about the y-axis and resembles two exponential curves, \(e^x\) for \( x > 0 \) and \(e^{-x}\) for \( x < 0 \), meeting at \(f(0)=1\).

Step 1: Understand the Function

The function given is \( f(x) = e^{|x|} \). This is an exponential function where the exponent is the absolute value of \( x \). Since absolute value is involved, the function is symmetric about the y-axis.

Step 2: Analyze the Function on x > 0

For \( x > 0 \), the absolute value \( |x| = x \), so the function simplifies to \( f(x) = e^x \). This is a standard exponential function that increases as \( x \) increases.

Step 3: Analyze the Function on x < 0

For \( x < 0 \), the absolute value \( |x| = -x \), so the function becomes \( f(x) = e^{-x} \). This is an exponential decay function, which decreases as \( x \) becomes more negative.

Step 4: Evaluate Key Points

Evaluate the function at key points: \( f(0) = e^{|0|} = 1 \), \( f(1) = e^1 = e \approx 2.718 \), \( f(-1) = e^1 = e \approx 2.718 \). This confirms the symmetry and helps plot the curve.

Step 5: Plot the Graph

Start plotting the graph. At \( x = 0 \), the function is \( 1 \). For \( x > 0 \), plot the standard exponential function, which climbs steeply to the right. For \( x < 0 \), plot the exponential decay function, mirroring the steep climb to the right.

Key Concepts

Absolute Value

In the context of the function \( f(x) = e^{|x|} \), absolute value is a key component. The absolute value of a number is the distance of that number from zero on the number line, without considering the direction. In simpler terms, it’s always positive or zero for a real number. This means that whether \( x \) is positive or negative, \( |x| \) is positive.

• For positive \( x \): \( |x| = x \)
• For negative \( x \): \( |x| = -x \)
• For zero \( x \): \( |x| = 0 \)

This concept is crucial for understanding how the function behaves on each side of the y-axis. Because of the absolute value, the values of the function on the negative side \( x < 0 \) mirror those on the positive side \( x > 0 \). This results in the symmetric nature of the function's graph.

Graphing Functions

Graphing the function \( f(x) = e^{|x|} \) involves understanding both exponential growth and decay due to the absolute value.

For \( x > 0 \), the function becomes \( e^x \), which is a typical exponential growth function. As \( x \) increases, the value of \( f(x) \) increases sharply. This part of the graph starts at the point (0,1) and climbs steeply upwards as \( x \) moves to the right.

For \( x < 0 \), the absolute value converts \( x \) into \( -x \). Here, the function becomes \( e^{-x} \), showing exponential decay as \( x \) becomes more negative. This portion of the graph also starts at (0,1) and increases steeply to the left, mirroring the exponential growth on the positive side.

The combination of these two behaviors due to the absolute value results in a continuous curve swooping upwards from both directions, forming a v-like shape.

Symmetry in Graphs

A graph is symmetric about the y-axis when every point \((x, y)\) on the graph corresponds to a point \((-x, y)\). This symmetry can be seen in functions that have forms such as \(e^{|x|}\).

In the function \( f(x) = e^{|x|} \), the presence of the absolute value introduces symmetry. Because \(|x|\) is the same whether \( x \) is positive or negative, the function computes the same \( y \)-value for both \( x \) and \(-x\).

• When \( x = 1 \), \( f(1) = e^1 = e \).
• When \( x = -1 \), \( f(-1) = e^1 = e \).

Both points are equidistant from the y-axis and equal in height, showing the mirror-like quality of the graph. This symmetry helps in easily plotting and predicting the shape of the function's graph across the y-axis.