Question

In Exercises \(39-42,\) draw the graph and determine the domain and range of the function. $$y=2 \ln (3-x)-4$$

Short answer

The graph of the function \( y = 2 \ln (3-x) - 4 \) is a decreasing curve starting at \( (3, \infty) \) and approaching the asymptote at \( x = 3 \). The domain of the function is \( (-\infty, 3) \) and the range of the function is all real numbers.

Step-by-step solution

Identify Key Points

The graph of \( y = 2 \ln (3-x) - 4 \) includes the points where y is undefined and where y is zero. For this function, y is undefined for \( x \geq 3 \) (as logarithm of negative number is undefined). The function has a vertical asymptote at \( x = 3 \). The function \( y = 2 \ln (3-x) - 4 = 0 \) when \( x = 3 - e^{2} \). So, \( x = 3 - e^{2} \) is another key point for this function.

Sketch the Graph

Plot the key points and asymptotes. Then, sketch the function passing through these points. The function is decreasing because of the negative sign associated with x inside the natural logarithm function.

Determine Domain and Range

The domain are all possible values for the independent variable, x. Since the logarithmic function is undefined for negative arguments, the domain for this function will include all values \( x < 3 \). The range is determined from the graph; it includes all real numbers as logarithmic functions can output any real numbers.