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The problem you are working with is based on the function \( h(x) = 2^x - 3 \), and I can see that there's a graph related to this function in the image you've uploaded. Let's break down what this function and graph represent.

### Function Analysis: \( h(x) = 2^x - 3 \)

1. **Exponential Nature**: The function \( h(x) = 2^x - 3 \) is an exponential function where the base of the exponent is 2. This means the graph will grow rapidly for larger values of \( x \).
   
2. **Vertical Shift**: The term "-3" represents a vertical shift of the graph downwards by 3 units. So, the graph of \( 2^x \) is shifted down by 3.

3. **Domain and Range**:
   - **Domain**: The domain of the function is all real numbers \( (-\infty, \infty) \), since exponential functions are defined for all \( x \).
   - **Range**: Since \( 2^x \) is always positive, \( 2^x - 3 \) will always be greater than -3. So the range is \( (-3, \infty) \).

4. **Intercept**:
   - **Y-intercept**: When \( x = 0 \), \( h(0) = 2^0 - 3 = 1 - 3 = -2 \). So, the y-intercept is at \( (0, -2) \).
   - **X-intercept**: We set \( h(x) = 0 \) to find the x-intercept:  
     \[
     2^x - 3 = 0 \implies 2^x = 3 \implies x = \log_2(3)
     \]
     Using a calculator, \( \log_2(3) \approx 1.585 \). Thus, the x-intercept is around \( (1.585, 0) \).

### Graph Interpretation:

From the graph, we can observe the following:
- The function has a y-intercept at \( (0, -2) \), as expected.
- The graph grows exponentially for positive \( x \) and approaches -3 from above as \( x \) becomes very negative.
  
### Key Points to Plot:
- **Asymptote**: There is a horizontal asymptote at \( y = -3 \).
- **Intercepts**: Y-intercept at \( (0, -2) \) and X-intercept at around \( (1.585, 0) \).
- **Growth**: For large positive \( x \), the function \( h(x) \) grows rapidly.

Would you like to go over any specific details or calculations? Here are some further questions based on this:

1. How does the base of an exponential function affect its growth rate?
2. What is the significance of vertical shifts in exponential functions?
3. How can you calculate the x-intercept of a logarithmic function like \( \log_2(3) \)?
4. How do horizontal asymptotes relate to the behavior of exponential functions as \( x \to -\infty \)?
5. What happens to the graph of \( h(x) = 2^x - 3 \) as \( x \) approaches very large positive numbers?

**Tip**: When working with exponential functions, always check for vertical shifts and asymptotes to understand the general behavior of the graph.

Résoudre et interpréter graphiquement la fonction h(x) = 2^x - 3.

Étudiez la fonction h(x) = 2^x - 3 avec une analyse complète du domaine, de l'asymptote, des intercepts et de son comportement exponentiel. Graphiquement, le déplacement vertical de -3 est visible avec une asymptote horizontale en y = -3.

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