Graph each function. $\text{ƒ}\left(x\right)=2^{(}-\mid x\mid )$

Verified step by step guidance

1. Identify the function: \( f(x) = 2^{-|x|} \). This is an exponential function with a negative exponent involving the absolute value of \( x \).
2. Understand the effect of the absolute value: The expression \( -|x| \) means that the exponent will always be non-positive, as \(|x|\) is always non-negative.
3. Consider the behavior of the function: As \( x \) moves away from zero in either direction, \(|x|\) increases, making \(-|x|\) more negative, which decreases \(2^{-|x|}\) towards zero.
4. Evaluate key points: Calculate \( f(x) \) at \( x = 0 \), \( x = 1 \), \( x = -1 \), \( x = 2 \), and \( x = -2 \) to understand the shape of the graph.
5. Sketch the graph: Plot the points calculated and note that the graph is symmetric about the y-axis due to the absolute value, and it approaches zero as \( x \) moves away from zero in both directions.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a^x, where 'a' is a positive constant. In the given function, f(x) = 2^(-|x|), the base is 2, and the exponent is the negative absolute value of x. This results in a function that decreases as x moves away from zero, reflecting the properties of exponential decay.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as |x| and is always non-negative. In the function f(x) = 2^(-|x|), the absolute value affects the exponent, ensuring that the output remains positive for all real values of x, which is crucial for graphing the function.

Graphing Techniques

Graphing techniques involve plotting points on a coordinate plane to visualize the behavior of a function. For f(x) = 2^(-|x|), one would calculate values for various x inputs, noting that the graph is symmetric about the y-axis due to the absolute value. Understanding how to interpret and sketch the graph is essential for analyzing the function's characteristics, such as its intercepts and asymptotic behavior.