Question
Find the range of the exponential function \( f(x)=4^{4} \). (Use symbolic notation and fractions where needed. Give your answers as intervals in the form \( (*, *) \). Use the symbol \( \infty \) for infinity, \( \cup \) for combining intervals, and an appropriate type of parenthesis " \( (", ") ", "[" \) or " " " depending on whether the interval is open or closed.) range of \( f: \)
Ask by Owen Phillips. in the United States
Jan 29,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The range of \( f(x) = 4^{x} \) is all positive real numbers, which is represented as \( (0, \infty) \).
Solution
Assuming the exponential function is \( f(x) = 4^{x} \), the range can be determined as follows:
An exponential function of the form \( f(x) = a^{x} \), where \( a > 0 \) and \( a \neq 1 \), always yields positive real numbers. Specifically:
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( 0 \) but never actually reaches \( 0 \).
- As \( x \) increases, \( f(x) \) grows without bound towards \( \infty \).
Therefore, the range of \( f(x) = 4^{x} \) is all positive real numbers.
**Range of \( f \):**
\[ (0, \infty) \]
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Simplify this solution The Deep Dive
The range of the function \( f(x) = 4^{4} \) is simply the single value it outputs since it's a constant function, and therefore does not take on a variety of values. The function evaluates to \( 256 \) for all \( x \). Thus, the range can be expressed as: \[ [256, 256] \] This indicates that the function outputs \( 256 \) exclusively.
