Question

Which one of the following functions best represent the graph as shown adjacent ?(A) \(\mathrm{f}(\mathrm{x})=\frac{1}{1+\mathrm{x}^{2}}\) (B) \(f(x)=\frac{1}{1+\sqrt{|x|}}\) (C) \(f(x)=e^{-j \mid}\) (D) \(f(x)=a^{x \mid}(a>1)\)

Short answer

Answer: (C) \(f(x) = e^{-|x|}\)

Step-by-step solution

Analyze Function A

For function \(A\), \(f(x) = \frac{1}{1 + x^2}\). Observe that this function is symmetric about the y-axis as the value of \(f(-x)\) will also be equal to the value of \(f(x)\). As x gets larger, in any direction, the denominator gets larger and \(f(x)\) approaches zero. So, this function starts at a peak at the origin and then tapers off as x gets larger.

Analyze Function B

For function \(B\), \(f(x) = \frac{1}{1 + \sqrt{|x|}}\). Since the absolute value of x is used, this function is also symmetric about the y-axis. However, as x gets larger in either direction, the denominator of the fraction also increases, and \(f(x)\) tends to zero. This function starts at a peak at the origin and then falls off as x gets larger, but more slowly than function A.

Analyze Function C

For function \(C\), \(f(x) = e^{-|x|}\). This function is also symmetric about the y-axis because of the absolute value of x in the exponent. As x gets larger in any direction, the exponent becomes more negative, which makes the entire function approach zero more quickly than the previous functions. This function also starts at a peak at the origin and then tapers off as x gets larger.

Analyze Function D

For function \(D\), \(f(x) = a^{|x|}(a>1)\). This function is not symmetric about the y-axis because the value of \(f(-x)\) will not always equal the value of \(f(x)\) unless a = 1, in which case the function is constant. This function does not start with a peak at the origin and is not symmetric about the y-axis, so it cannot represent the given graph.

Compare Functions to Given Graph

From the analysis of individual functions, we know that all the functions except function D are symmetric about the y-axis, and that they all approach zero as x gets further away from the origin. The difference between functions A, B, and C is in how quickly they approach zero. Based on the graph properties, it would be wise to choose the function that best matches the rate at which the graph tapers off as x increases. After taking into account the analysis of each function and comparing it to the given graph's behavior, the best choice for the given problem would be: (C) \(f(x) = e^{-|x|}\)