Question

Sketch the graph of the function. $$ f(x)=3^{-x^{2}} $$

Short answer

The graph of the function \( f(x) = 3^{-x^{2}} \) is symmetric with respect to the y-axis, has a y-intercept at point (0,1), no x-intercepts, a horizontal asymptote at \( y = 0 \), and has its domain as all real numbers and its range as all positive real numbers.

Step-by-step solution

Determine the Symmetry

Firstly, ascertain if the function is symmetric. A function \( f(x) \) is even if \( f(x) = f(-x) \) for any \( x \) in the function's domain. A function is odd if \( -f(x) = f(-x) \). In this case, \( f(x) = f(-x) \), and therefore, the given function is an even function. This means that the graph will be symmetric with respect to the y-axis.

Find the Intercepts

Intercepts are the points where the graph hits the x-axis (x-intercepts) and the y-axis (y-intercepts). For the y-intercepts, set \( x = 0 \) in the function: \( f(0) = 3^{-(0)^{2}} = 3^{0} = 1 \). So, the y-intercept is at the point (0,1). The x-intercepts are the 'x' values when \( f(x) = 0 \). However, since the value of an exponential function is always greater than zero, there are no x-intercepts in this case.

Check for Asymptotes

Horizontal asymptotes are lines that the graph approaches but never touches. For the function \( f(x) = 3^{-x^{2}} \), as \( x \) tends to negative or positive infinity, \( f(x) \) approaches 0. So, the x-axis, i.e., the line \( y = 0 \), is the horizontal asymptote here. There are no vertical asymptotes, which would be values of \( x \) for which \( f(x) \) tends to positive or negative infinity.

Determine the Domain and Range

The domain of a function is the set of 'x' values for which the function is defined. Since \( f(x) = 3^{-x^{2}} \) for any real x, the domain is all real numbers. The range is the set of all 'y' values that the function can take. As earlier explained, \( f(x) \) is always greater than zero (but never equals zero), so the range is all positive real numbers.

Sketch the Graph

With all the information gathered, the graph can be sketched. It is symmetric about the y-axis, touches the y-axis at point (0,1), falls towards but never touches the line \( y = 0 \) (the x-axis). The function is defined for all real x-values, but only takes positive y-values, making it above the x-axis at all times.