State the range for the function  f(x) = 2^12x − 5: 1
(5, ∞)
(-∞, ∞)
(-5, ∞)
(0, ∞)

To find the range of the function \( f(x) = 2^{12x} - 5 \), let's break it down:

1. **Base Function**: The expression \( 2^{12x} \) is an exponential function, where \( 2^y > 0 \) for all real values of \( y \). The exponential function grows rapidly but is always positive, meaning \( 2^{12x} \in (0, \infty) \).

2. **Shifted Function**: The function is then shifted down by 5 units. Therefore, the smallest value of \( 2^{12x} \) (which is 0) will map to \( -5 \), and as \( 2^{12x} \) grows, the function will approach \( \infty \).

Thus, the range of \( f(x) = 2^{12x} - 5 \) is:
\[
\boxed{(-5, \infty)}
\]

Would you like a deeper explanation of how this works? Here are some related questions:

1. What is the general form of an exponential function?
2. How does shifting a function vertically affect its range?
3. Can an exponential function ever equal zero?
4. What are the effects of horizontal scaling in \( 2^{12x} \)?
5. How do you find the domain and range of other types of functions?

**Tip:** When working with exponential functions, keep in mind that their range is always positive, and any vertical shifts directly affect the minimum or maximum values.

State the range for the function f(x) = 2^12x − 5: 1 (5, ∞) (-∞, ∞) (-5, ∞) (0, ∞)

Learn how to find the range of the exponential function f(x) = 2^12x − 5. This problem explores key concepts such as the behavior of exponential functions and how vertical shifts affect the range. Suitable for high school students in Grades 10-12.

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