https://r2.erweima.ai/mathbot/image/1727044500731o5bqzyna.jpg

Let's tackle the exercises one by one from the provided image.

### **Exercise 16**  
For each function, we need to analyze the following aspects:
- **Domain**: The set of \( x \)-values for which the function is defined.
- **Zeros**: Values of \( x \) where the function equals zero.
- **Y-intercept**: The point where the function crosses the y-axis (i.e., \( f(0) \)).
- **Asymptotes**: Horizontal or vertical lines that the function approaches but never reaches.
- **Sign of the function**: Where the function is positive or negative.
- **Approximate Graph**: A general sketch of the graph.

#### a. \( f(x) = \left( \frac{1}{5} \right)^x - 2 \)
- **Domain**: \( (-\infty, \infty) \) (exponential functions are defined for all real \( x \)).
- **Zeros**: Solve \( \left( \frac{1}{5} \right)^x - 2 = 0 \) ⇒ \( \left( \frac{1}{5} \right)^x = 2 \) ⇒ \( x = \log_{\frac{1}{5}} 2 \).
- **Y-intercept**: \( f(0) = \left( \frac{1}{5} \right)^0 - 2 = 1 - 2 = -1 \).
- **Asymptotes**: Horizontal asymptote at \( y = -2 \).
- **Sign**: Positive for values of \( x \) where \( \left( \frac{1}{5} \right)^x > 2 \), negative otherwise.
- **Graph**: The graph is an exponential decay shifted down by 2 units, crossing the y-axis at \( (0, -1) \).

#### b. \( f(x) = -2 \cdot (5)^x + 3 \)
- **Domain**: \( (-\infty, \infty) \).
- **Zeros**: Solve \( -2 \cdot 5^x + 3 = 0 \) ⇒ \( 5^x = \frac{3}{2} \) ⇒ \( x = \log_5 \left( \frac{3}{2} \right) \).
- **Y-intercept**: \( f(0) = -2 \cdot 5^0 + 3 = -2 + 3 = 1 \).
- **Asymptotes**: None, as it's an exponential function.
- **Sign**: Positive for values of \( x \) where \( -2 \cdot 5^x + 3 > 0 \), negative otherwise.
- **Graph**: Exponential growth reflected vertically and shifted up by 3 units.

#### c. \( f(x) = \ln(x + 2) \)
- **Domain**: \( x + 2 > 0 \) ⇒ \( x > -2 \) (Domain: \( (-2, \infty) \)).
- **Zeros**: Solve \( \ln(x + 2) = 0 \) ⇒ \( x + 2 = 1 \) ⇒ \( x = -1 \).
- **Y-intercept**: \( f(0) = \ln(0 + 2) = \ln(2) \).
- **Asymptotes**: Vertical asymptote at \( x = -2 \).
- **Sign**: Positive for \( x > -1 \), negative for \( -2 < x < -1 \).
- **Graph**: Logarithmic graph shifted to the left by 2 units.

#### d. \( f(x) = \ln(x) - 3 \)
- **Domain**: \( x > 0 \).
- **Zeros**: Solve \( \ln(x) - 3 = 0 \) ⇒ \( \ln(x) = 3 \) ⇒ \( x = e^3 \).
- **Y-intercept**: Undefined (since \( \ln(0) \) is undefined).
- **Asymptotes**: Vertical asymptote at \( x = 0 \).
- **Sign**: Positive for \( x > e^3 \), negative for \( 0 < x < e^3 \).
- **Graph**: Logarithmic curve shifted down by 3 units.

#### e. \( f(x) = \sqrt{-x + 3} \)
- **Domain**: \( -x + 3 \geq 0 \) ⇒ \( x \leq 3 \).
- **Zeros**: Solve \( \sqrt{-x + 3} = 0 \) ⇒ \( -x + 3 = 0 \) ⇒ \( x = 3 \).
- **Y-intercept**: \( f(0) = \sqrt{3} \).
- **Asymptotes**: None.
- **Sign**: Non-negative, as square root functions are always positive or zero.
- **Graph**: A square root graph, reflected horizontally.

#### f. \( f(x) = \sqrt[3]{x - 9} + 4 \)
- **Domain**: \( (-\infty, \infty) \) (cube roots are defined for all real \( x \)).
- **Zeros**: Solve \( \sqrt[3]{x - 9} + 4 = 0 \) ⇒ \( \sqrt[3]{x - 9} = -4 \) ⇒ \( x - 9 = (-4)^3 = -64 \) ⇒ \( x = -55 \).
- **Y-intercept**: \( f(0) = \sqrt[3]{0 - 9} + 4 = \sqrt[3]{-9} + 4 \).
- **Asymptotes**: None.
- **Sign**: Can be both positive and negative depending on the cube root values.
- **Graph**: Cube root shifted to the right and up.

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### **Exercise 17**
Graph each piecewise function approximately and analyze domain, zeros, and y-intercept.

#### a. \( f(x) = \frac{x^2 + 1}{x} \) for \( x > 2 \), and \( f(x) = x \) for \( x \leq 2 \)
- **Domain**: \( (-\infty, \infty) \) excluding \( x = 0 \).
- **Zeros**: Solve each piece for where \( f(x) = 0 \). For \( x > 2 \), no zeros. For \( x \leq 2 \), zero at \( x = 0 \).
- **Y-intercept**: \( f(0) \) is undefined.
- **Graph**: Two pieces — quadratic and linear.

#### b. \( f(x) = \{-3 \text{ if } x \leq 1; x + 5 \text{ if } x > 1 \} \)
- **Domain**: \( (-\infty, \infty) \).
- **Zeros**: \( x + 5 = 0 \) gives \( x = -5 \).
- **Y-intercept**: \( f(0) = -3 \).

#### c. \( f(x) = \{-1 \text{ if } x < 0; 1 \text{ if } x = 0; x^2 - 1 \text{ if } x > 0 \} \)
- **Domain**: \( (-\infty, \infty) \).

#### d. \( f(x) = \{-x^2 + 4 \text{ if } x > 1; x \text{ if } x \leq 1 \} \)
- **Domain**: \( (-\infty, \infty) \).

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Would you like me to continue graphing, or are there specific parts you'd like more explanation on? 

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### 5 Related Questions:
1. How can we graph exponential functions quickly?
2. What is the domain of logarithmic functions and how to find it?
3. What are common transformations for square root functions?
4. How do asymptotes impact the behavior of graphs?
5. How do piecewise functions differ from standard functions?

### Tip:
For exponential functions, look at shifts and stretches to determine their key points and behavior.

Para las siguientes funciones analiza: dominio, ceros, ordenada al origen, asíntotas, signos y gráfica aproximada.

Learn how to analyze and graph various functions including exponential functions, logarithmic functions, square root functions, and piecewise functions. This includes finding the domain, zeros, y-intercept, asymptotes, and an approximate graph of each function. Suitable for high school students.

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