Understanding Function Transformations and Asymptotes

The video tutorial explains how to analyze the function f(x) = 1/(x+4) - 5 by using transformations of the basic function g(x) = 1/x. It covers the domain, range, and asymptotes of both functions. The tutorial demonstrates how to apply horizontal and vertical shifts to determine the new asymptotes and graph of f(x). It concludes with identifying the domain and range of the transformed function.

What is the domain of the basic function g(x) = 1/x?

• All real numbers except x = 0

• All negative numbers

• All real numbers

• All positive numbers

What is the vertical asymptote of the function g(x) = 1/x?

• x = 0

• x = -1

• x = 1

• x = 2

How does the transformation g(x + 4) affect the graph of g(x)?

• Shifts the graph up by 4 units

• Shifts the graph left by 4 units

• Shifts the graph right by 4 units

• Shifts the graph down by 4 units

What happens to the horizontal asymptote when the graph is shifted down by 5 units?

• It shifts down by 5 units

• It remains unchanged

• It shifts left by 5 units

• It shifts up by 5 units

If a is positive, how is the graph of g(x) = 1/x affected?

• Shifted right by a units

• Shifted left by a units

• Shifted up by a units

• Shifted down by a units

What is the new vertical asymptote of f(x) after shifting g(x) left by 4 units?

• x = -5

• x = 0

• x = -4

• x = 4

What is the horizontal asymptote of f(x) after shifting g(x) down by 5 units?

• y = 0

• y = -4

• y = -5

• y = 5

What is the domain of the function f(x) after the transformations?

• All negative numbers

• All positive numbers

• All real numbers except x = -4

• All real numbers

What is the range of the function f(x) after the transformations?

• All real numbers

• All real numbers except y = -5

• All negative numbers

• All positive numbers

Why is x = -4 not included in the domain of f(x)?

• Because it results in a negative output

• Because it results in division by zero

• Because it results in a positive output

• Because it results in a zero output