Let's analyze the function $f(x) = 2(2^x) - 4$ and fill in the blanks based on its features.

Step-by-Step Analysis

1. Type of Function: $f(x) = 2 \cdot 2^x - 4$ is an exponential function because the variable $x$ is in the exponent.

2. Horizontal Asymptote: For exponential functions of the form $f(x) = a \cdot b^x + c$, the horizontal asymptote is $y = c$. In this case, the horizontal asymptote is: $y = -4$

3. Range: The function $f(x)$ approaches but never reaches $-4$ as $x \to -\infty$, and it goes to $+\infty$ as $x \to +\infty$. Thus, the range of $f(x)$ is: $(-4, \infty)$

4. Domain: Exponential functions are defined for all real numbers. Therefore, the domain is: $(-\infty, \infty)$

5. End Behavior:

   • Left Side ($x \to -\infty$): As $x$ goes to negative infinity, $2^x \to 0$, so $f(x)$ approaches the horizontal asymptote $y = -4$ from above. Thus, $f(x) \to -4^+$.
   • Right Side ($x \to +\infty$): As $x$ increases, $2^x \to +\infty$, so $f(x) \to +\infty$.

Filling the Blanks

Based on the analysis:

• The function $f(x)$ is an exponential function.
• The asymptote is horizontal at $y = -4$.
• The range of the function is $(-4, \infty)$.
• The function is increasing on its domain of $(-\infty, \infty)$.
• The end behavior on the left side is as $x \to -\infty$, $f(x) \to -4^+$.
• The end behavior on the right side is as $x \to +\infty$, $f(x) \to +\infty$.

Final Answer

"The function $f(x)$ is an exponential function with a horizontal asymptote of $y = -4$. The range of the function is $(-4, \infty)$, and it is increasing on its domain of $(-\infty, \infty)$. The end behavior on the left side is as $x \to -\infty$, $f(x) \to -4^+$, and the end behavior on the right side is as $x \to +\infty$, $f(x) \to +\infty$."

Would you like further details on any part of the solution?

Related Questions:

1. What other types of transformations can affect the graph of an exponential function?
2. How would the range change if the function had a vertical shift upwards?
3. What happens to the asymptote if we change the constant outside the exponent?
4. How do changes in the base of the exponent affect the function's growth rate?
5. How does a reflection over the x-axis affect the end behavior of an exponential function?

Tip: Remember, the horizontal asymptote of an exponential function typically reflects the vertical shift in the function.