https://file.aiquickdraw.com/mathbot/file/1726701104178inwjlq4r.jpg

The given function is defined as:

\[
f(x) = \left(\frac{1}{3}\right)^{-(x+3)}
\]

This can be rewritten as:

\[
f(x) = 3^{x+3}
\]

This is an exponential function with base 3, shifted horizontally and vertically. Now, let's analyze the characteristics of this function step by step.

### (i) Horizontal Asymptote:
For any exponential function of the form \( f(x) = 3^{x+3} \), the horizontal asymptote occurs as \( x \to -\infty \). The function tends to 0, meaning the horizontal asymptote is:

\[
y = 0
\]

### (ii) Range:
Since exponential functions are always positive, the range will be:

\[
(0, \infty)
\]

### (iii) y-intercept:
To find the y-intercept, we evaluate \( f(0) \):

\[
f(0) = 3^{0+3} = 3^3 = 27
\]

So the y-intercept is:

\[
(0, 27)
\]

### (iv) Letter Corresponding to the Graph:
The function \( f(x) = 3^{x+3} \) represents an increasing exponential function with a steep rise and an asymptote at \( y = 0 \). Graph C matches this description, where the curve rapidly increases as \( x \) grows.

### Final Answers:
1. **Horizontal Asymptote**: \( y = 0 \)
2. **Range**: \( (0, \infty) \)
3. **y-intercept**: \( 27 \)
4. **Corresponding Graph**: Graph C

Let me know if you'd like further details or explanations on any part of this solution!

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Here are 5 related questions to deepen your understanding:
1. How does changing the base of an exponential function affect its graph?
2. What transformations occur when adding or subtracting values inside the exponent?
3. What is the significance of the horizontal asymptote in exponential functions?
4. How would the graph change if the base of the function were a fraction instead of 3?
5. How do you determine the domain of an exponential function?

**Tip**: For exponential functions, remember that the horizontal asymptote is influenced by the vertical shift, but the base determines how quickly the function grows or decays.

Let f be the exponential function defined by f(x) = (1/3)^(-(x+3)) and determine the following information based on f: (i) Horizontal Asymptote: (ii) Range: (iii) y-intercept: (iv) Letter Corresponding to Graph:

Solve the exponential function f(x) = (1/3)^(-(x+3)) by determining the horizontal asymptote, range, y-intercept, and identifying the corresponding graph. This exercise covers key concepts in exponential functions, including graph transformations and asymptotes. Suitable for Grades 10-12.

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