Find the domain and range

1. y = - 2 ^ (5x - 1)

2. y = 5 ^ (3x + 1) - 7

3. y = - 7 ^ (2x - 3) + 5

Let's analyze each function individually to determine its domain and range:

### 1. \( y = -2^{(5x - 1)} \)

- **Domain:** Exponential functions are defined for all real numbers. Therefore, the domain is:
  \[
  \text{Domain: } (-\infty, \infty)
  \]

- **Range:** The function \( 2^{(5x - 1)} \) is an exponential function that grows larger as \( x \) increases and approaches 0 as \( x \) decreases. Since the base of the exponential is positive and the coefficient is negative, the function will always be negative. The output values will approach 0 from below, but will never reach it.

  \[
  \text{Range: } (-\infty, 0)
  \]

---

### 2. \( y = 5^{(3x + 1)} - 7 \)

- **Domain:** Again, exponential functions are defined for all real numbers. Therefore, the domain is:
  \[
  \text{Domain: } (-\infty, \infty)
  \]

- **Range:** The base function \( 5^{(3x + 1)} \) is always positive and grows larger as \( x \) increases. Subtracting 7 shifts the entire function down by 7 units. The minimum value of \( y \) occurs when \( x \to -\infty \), where \( 5^{(3x + 1)} \to 0 \). Therefore, the function will approach \( -7 \) but never reach it, and it will increase without bound as \( x \to \infty \).

  \[
  \text{Range: } (-7, \infty)
  \]

---

### 3. \( y = -7^{(2x - 3)} + 5 \)

- **Domain:** Similar to the other cases, exponential functions are defined for all real numbers. Therefore, the domain is:
  \[
  \text{Domain: } (-\infty, \infty)
  \]

- **Range:** The base function \( 7^{(2x - 3)} \) is always positive and grows as \( x \) increases. The negative sign flips the function, and adding 5 shifts the function upward by 5 units. The largest value of \( y \) occurs when \( 7^{(2x - 3)} \to 0 \), which gives \( y = 5 \). As \( x \to \infty \), \( 7^{(2x - 3)} \) becomes very large, and \( y \) decreases without bound.

  \[
  \text{Range: } (-\infty, 5)
  \]

---

### Summary:

1. \( y = -2^{(5x - 1)} \)
   - Domain: \( (-\infty, \infty) \)
   - Range: \( (-\infty, 0) \)

2. \( y = 5^{(3x + 1)} - 7 \)
   - Domain: \( (-\infty, \infty) \)
   - Range: \( (-7, \infty) \)

3. \( y = -7^{(2x - 3)} + 5 \)
   - Domain: \( (-\infty, \infty) \)
   - Range: \( (-\infty, 5) \)

---

Do you have any questions or would you like further details on any part of this? 

Here are five related questions you might explore:

1. How does the base of an exponential function affect its growth rate?
2. How would the domain and range change if the base were negative?
3. What happens to the range if the exponent contains an absolute value term?
4. How would the graph of these functions look, and what are the key features?
5. How do transformations (like vertical shifts) affect the domain and range?

**Tip:** When working with exponential functions, keep in mind that any negative exponent will result in a reciprocal, making the function approach zero as \( x \) decreases.

Find the domain and range for the following functions: 1. y = -2^(5x - 1), 2. y = 5^(3x + 1) - 7, 3. y = -7^(2x - 3) + 5

Explore how to find the domain and range of three exponential functions: -2^(5x - 1), 5^(3x + 1) - 7, and -7^(2x - 3) + 5. This step-by-step guide covers exponential function properties, including shifts and reflections. Suitable for high school students in Grades 10-12.

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