Question

several ordered pairs from a continuous exponential function are shown in the table.

x	y
1	5
2	6.25
3	7.8125

what are the domain and range of the function?

• the domain is the set of integers, and the range is ( y > 4 ).
• the domain is the set of integers, and the range is ( y > 0 ).
• the domain is the set of real numbers, and the range is ( y > 0 ).
• the domain is the set of real numbers, and the range is ( y > 4 ).

Explanation:

1. Domain of Exponential Function: Exponential functions are defined for all real numbers. The table shows integer values of \( x \), but the function is continuous (as stated), so \( x \) can be any real number (e.g., fractions, decimals), not just integers.
2. Range of Exponential Function: The general form of a continuous exponential function is \( y = ab^x + k \) (for growth/decay). From the table, when \( x = 0 \), \( y = 4 \), and as \( x \) changes, \( y \) is always positive (since exponential functions with \( b>0, b

eq1 \) and no vertical shift below 0 will have \( y>0 \) if the horizontal asymptote is \( y = 0 \); here, the values start at 4 and increase, so \( y>0 \) (the horizontal asymptote for this function is \( y = 0 \), so \( y \) approaches 0 but never reaches it, and is always positive).

Now, analyze the options:

• First option: Domain as integers is wrong (exponential is continuous, domain is real numbers); range \( y>4 \) is wrong (as \( x \) could approach \( -\infty \), \( y \) approaches 0, not 4).
• Second option: Domain as integers is wrong.
• Third option: Domain is real numbers (correct, since exponential functions are defined for all real \( x \)) and range \( y>0 \) (correct, as exponential functions with positive base have range \( y>0 \) when there's no vertical shift below 0, and here the function grows from near 0 to 4 and beyond, so \( y>0 \)).
• Fourth option: Range \( y>4 \) is wrong (as \( y \) can be between 0 and 4 as \( x \) decreases).

Answer:

The domain is the set of real numbers, and the range is \( y > 0 \). (Corresponding to the third option: "The domain is the set of real numbers, and the range is \( y > 0 \).")