Understanding the Function ( g(x) = e^x - x - 1 )

 

The function ( g(x) = e^x - x - 1 ) is an interesting mathematical expression that involves the natural exponential function. Let's break it down and explore its key characteristics:

Components of the Function

1. Exponential Term:

   • ( e^x ) is the natural exponential function, where ( e ) is approximately 2.71828. It is a fundamental mathematical constant and the base of natural logarithms.

   • The exponential function grows rapidly as ( x ) increases and approaches zero as ( x ) becomes negative.

2. Linear Term:

   • The term (-x) is a linear function with a slope of (-1). It represents a straight line that decreases as ( x ) increases.

3. Constant Term:

   • The constant (-1) shifts the graph of the function downward by one unit.

Key Properties

Domain and Range

• Domain: The domain of ( g(x) ) is all real numbers ((-\infty, \infty)), as there are no restrictions on the values ( x ) can take in the exponential, linear, or constant terms.

• Range: The range of ( g(x) ) is also all real numbers because the exponential function ( e^x ) can take on any positive value, and the linear term (-x) and constant term (-1) allow for negative values as well.

Intercepts

• Y-Intercept: The ( y )-intercept occurs when ( x = 0 ). Substituting ( x = 0 ) into the function gives: [ g(0) = e^0 - 0 - 1 = 1 - 1 = 0 ] Hence, the graph passes through the origin (0,0).

Behavior and Symmetry

• Asymptotic Behavior:

  • As ( x ) approaches negative infinity, ( e^x ) approaches zero, making ( g(x) \approx -x - 1 ). Hence, the function decreases without bound.

  • As ( x ) approaches positive infinity, ( e^x ) dominates both (-x) and (-1), causing the function to increase without bound.

• Symmetry: The function ( g(x) = e^x - x - 1 ) is neither even nor odd, as it does not exhibit symmetry about the ( y )-axis or the origin.

Graphical Representation

The graph of ( g(x) = e^x - x - 1 ) will display a continuous curve that passes through the origin and exhibits exponential growth as ( x ) increases. The curve will be above the line ( y = -x - 1 ) for negative ( x ) and will surpass the line ( y = e^x ) for positive ( x ).

Derivative and Critical Points

To find critical points and analyze the behavior of the function, we can calculate the derivative:

[ g'(x) = \frac{d}{dx}(e^x - x - 1) = e^x - 1 ]

Setting the derivative equal to zero to find critical points:

[ e^x - 1 = 0 \Rightarrow e^x = 1 \Rightarrow x = 0 ]

This indicates a critical point at ( x = 0 ). The second derivative test or further analysis would be required to classify the nature of this critical point.

In summary, the function ( g(x) = e^x - x - 1 ) is a dynamic and complex function that illustrates the interplay between exponential growth and linear decay, with a noticeable impact at the origin.