The given function is $f(x) = 2 \left(\frac{1}{2}\right)^x - 5$, which is an exponential decay function. Let's analyze its features based on the graph provided:

1. Type of Function:

   • This is an exponential decay function since the base of the exponential, $\frac{1}{2}$, is less than 1.

2. Asymptote:

   • The graph shows a horizontal asymptote at $y = -5$. This is because, as $x \to \infty$, the exponential term $2 \left(\frac{1}{2}\right)^x$ approaches 0, leaving $y = -5$.

3. Range:

   • The range of the function is $(-5, \infty)$, because the graph never goes below $y = -5$, but it increases without bound as $x \to -\infty$.

4. Domain:

   • The domain is all real numbers, $(-\infty, \infty)$, since exponential functions are defined for all real $x$.

5. End Behavior:

   • As $x \to -\infty$, the function grows very large, so $f(x) \to \infty$.
   • As $x \to \infty$, the function approaches the asymptote, so $f(x) \to -5$.

Filling in the blanks:

• The function $f(x)$ is an exponential decay function with a horizontal asymptote of $y = -5$.
• The range of the function is $(-5, \infty)$, and it is increasing on its domain of $(-\infty, \infty)$.
• The end behavior on the left side is as $x \to -\infty$, $f(x) \to \infty$.
• The end behavior on the right side is as $x \to \infty$, $f(x) \to -5$.

Would you like further details on this?
Here are some related questions to expand your understanding:

1. How do we determine the base of an exponential function from its equation?
2. What conditions affect the steepness of an exponential decay graph?
3. How can you find the horizontal asymptote for any given exponential function?
4. What is the significance of the end behavior in understanding the graph of a function?
5. Can the domain of an exponential function ever be restricted, and why?

Tip: In exponential decay functions like this, the smaller the base (closer to 0), the faster the function decays.