Questions: Use transformations to graph the function. Determine the domain, range, horizontal asymptote, and y-intercept of the function. f(x) = 5^x + 1 Use the graphing tool to graph the function. Click to enlarge graph (For any answer boxes shown with the grapher, type an exact answer.) What is the domain of f(x) = 5^x + 1? (Type your answer in interval notation.) What is the range of f(x) = 5^x + 1? (Type your answer in interval notation.) Determine the horizontal asymptote of f(x) = 5^x + 1. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

Solution Steps

The domain of a function is the set of all possible input values (x-values) that the function can accept. For the function \( f(x) = 5^x + 1 \), there are no restrictions on the values of \( x \). Therefore, the domain is all real numbers.

Domain: \( (-\infty, \infty) \)

The range of a function is the set of all possible output values (y-values). For the function \( f(x) = 5^x + 1 \), the term \( 5^x \) is always positive and approaches 0 as \( x \) approaches negative infinity. Therefore, \( 5^x + 1 \) will always be greater than 1.

Range: \( (1, \infty) \)

A horizontal asymptote is a horizontal line that the graph of the function approaches as \( x \) approaches positive or negative infinity. For the function \( f(x) = 5^x + 1 \), as \( x \) approaches negative infinity, \( 5^x \) approaches 0, and thus \( f(x) \) approaches 1.

Horizontal Asymptote: \( y = 1 \)

Final Answer