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

The given function is:

\[
f(x) = 2^{12x} - 5
\]

To find the range, let's analyze it step by step:

1. **Base Function**: The function \( g(x) = 2^{12x} \) is an exponential function, and exponential functions always have positive outputs, with \( g(x) \) taking values in the interval \( (0, \infty) \).

2. **Subtracting 5**: The transformation \( f(x) = 2^{12x} - 5 \) shifts the graph of \( g(x) = 2^{12x} \) downward by 5 units. This means that instead of ranging from \( (0, \infty) \), the function now ranges from \( (-5, \infty) \).

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

\[
\boxed{(-5, \infty)}
\]

Let me know if you'd like further details or have any questions.

Here are 5 related questions to explore:

1. What is the domain of \( f(x) = 2^{12x} - 5 \)?
2. How does changing the exponent \( 12x \) affect the graph of \( f(x) \)?
3. What is the horizontal asymptote of the function \( f(x) = 2^{12x} - 5 \)?
4. How does the function \( f(x) = 2^{kx} - 5 \) behave for different values of \( k \)?
5. What would happen to the range if the function were \( f(x) = 2^{12x} + 5 \)

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

Understand how to find the range of the exponential function f(x) = 2^{12x} - 5. This problem covers essential concepts in exponential functions and function transformations, suitable for students in Grades 9-12.

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