Question

use the graph of y = e^x and transformations to sketch the exponential function f(x) = e^x - 1. determine the domain and range. also, determine the y-intercept, and find the equation of the horizontal asymptote. (for any answer boxes shown with the grapher, type an exact answer.) what is the domain of f(x) = e^x - 1? (type your answer in interval notation.) what is the range of f(x) = e^x - 1? (type your answer in interval notation.) what is the y-intercept of f(x) = e^x - 1? (type an integer or a simplified fraction.) what is the horizontal asymptote of f(x) = e^x - 1? (type an equation.)

Explanation:

Step1: Determine the domain of \( f(x) = e^x - 1 \)

The function \( y = e^x \) has a domain of all real numbers because the exponential function \( e^x \) is defined for any real - valued input \( x \). When we perform a vertical shift (subtracting 1 from \( e^x \)) to get \( f(x)=e^x - 1 \), the domain is not affected by vertical shifts. So the domain of \( f(x)=e^x - 1 \) is all real numbers. In interval notation, this is \( (-\infty,\infty) \).

Step2: Determine the range of \( f(x) = e^x - 1 \)

The range of \( y = e^x \) is \( (0,\infty) \) because \( e^x>0 \) for all real \( x \). For the function \( f(x)=e^x - 1 \), we are taking the graph of \( y = e^x \) and shifting it down by 1 unit. If \( y = e^x \) has values greater than 0, then \( y=e^x - 1 \) will have values greater than \( 0 - 1=-1 \). So the range of \( f(x)=e^x - 1 \) is \( (-1,\infty) \).

Step3: Determine the y - intercept of \( f(x) = e^x - 1 \)

The y - intercept of a function is found by setting \( x = 0 \) and evaluating the function. Substitute \( x = 0 \) into \( f(x)=e^x - 1 \):
\( f(0)=e^0 - 1 \)
Since \( e^0 = 1 \) (by the property of exponential functions where any non - zero number to the power of 0 is 1), then \( f(0)=1 - 1 = 0 \).

Step4: Determine the horizontal asymptote of \( f(x) = e^x - 1 \)

The horizontal asymptote of \( y = e^x \) is \( y = 0 \) (as \( x
ightarrow-\infty \), \( e^x
ightarrow0 \)). When we shift the graph of \( y = e^x \) down by 1 unit to get \( f(x)=e^x - 1 \), the horizontal asymptote also shifts down by 1 unit. So the horizontal asymptote of \( f(x)=e^x - 1 \) is \( y=-1 \).

Answer:

s:

• Domain: \( (-\infty,\infty) \)
• Range: \( (-1,\infty) \)
• y - intercept: \( 0 \)
• Horizontal asymptote: \( y = - 1 \)