Questions: Use the graph of y=e^x and transformations to sketch the exponential function f(x)=e^-x+5. Determine the domain and range. Also, determine the y-intercept, and find the equation of the horizontal asymptote.

Solution Steps

The domain of \( f(x) = e^{-x} + 5 \) is all real numbers, \( (-\infty, \infty) \).

The range of \( f(x) = e^{-x} + 5 \) is \( (5, \infty) \) because \( e^{-x} \) is always positive and approaches 0 as \( x \) approaches infinity.

To find the y-intercept, set \( x = 0 \): \[ f(0) = e^{-0} + 5 = 1 + 5 = 6 \] So, the y-intercept is \( (0, 6) \).

As \( x \) approaches infinity, \( e^{-x} \) approaches 0. Therefore, the horizontal asymptote is: \[ y = 5 \]

Final Answer