To analyze the function $f(x) = -2(2^x) + 5$, we'll look at several features including asymptotes, range, domain, and end behavior.

1. Type of Function:

This is an exponential function with a transformation.

2. Domain:

The domain of an exponential function is all real numbers, as there are no restrictions on the values of $x$.

• Domain: $(-\infty, \infty)$

3. Range:

The range of the function depends on the vertical transformations. The basic exponential function $2^x$ has a range of $(0, \infty)$. However, multiplying by -2 flips it and stretches it, and adding 5 shifts the entire graph upwards by 5 units.

• Range: $(-\infty, 5)$

4. Horizontal Asymptote:

For the function $f(x) = -2(2^x) + 5$, the horizontal asymptote occurs where the exponential part diminishes (as $x$ goes to negative infinity).

• Asymptote: $y = 5$

5. End Behavior:

• As $x \to -\infty$: $2^x \to 0$, so $f(x) \to -2(0) + 5 = 5$. Thus, the function approaches the horizontal asymptote from below.
• As $x \to \infty$: $2^x$ grows very large, so $f(x) = -2(2^x) + 5$ becomes very negative. Therefore, $f(x) \to -\infty$.

6. Summary of Features:

• Domain: $(-\infty, \infty)$
• Range: $(-\infty, 5)$
• Horizontal Asymptote: $y = 5$
• End Behavior:
  • As $x \to -\infty$, $f(x) \to 5$
  • As $x \to \infty$, $f(x) \to -\infty$

Would you like more details or have any specific questions?

Related Questions:

1. How do transformations affect the graph of an exponential function?
2. What is the significance of the horizontal asymptote in this function?
3. How do you determine the range of transformed exponential functions?
4. What is the general form of an exponential function and how does it differ from this specific one?
5. How do you find the intercepts of this function?

Tip: To visualize exponential functions and their transformations, graphing can be very helpful. Consider using a graphing calculator or software to see how shifts and stretches affect the function.