https://file.aiquickdraw.com/mathbot/file/17267610127252in394h8.jpg

Let's tackle each problem from the image one by one.

### 1. **Domain of the Function: \( f(x) = \sqrt{x - 19} \)**

For a square root function, the expression inside the square root must be non-negative because the square root of a negative number is not defined in the set of real numbers.

\[
x - 19 \geq 0
\]
Solving for \( x \):

\[
x \geq 19
\]

Therefore, the domain of \( f(x) = \sqrt{x - 19} \) is all \( x \geq 19 \). In interval notation, the domain is:

\[
[19, \infty)
\]

---

### 2. **Solve \( 5e^{t - 2} = 5 \) for \( t \)**

We start by dividing both sides by 5:

\[
e^{t - 2} = 1
\]

Next, take the natural logarithm (ln) of both sides:

\[
\ln(e^{t - 2}) = \ln(1)
\]

Since \( \ln(1) = 0 \) and \( \ln(e^x) = x \), we have:

\[
t - 2 = 0
\]

Solving for \( t \):

\[
t = 2
\]

---

### 3. **Domain of the Function: \( f(x) = \frac{4x + 6}{x - 3} \)**

For a rational function, the denominator cannot be zero, because division by zero is undefined.

Set the denominator equal to zero and solve for \( x \):

\[
x - 3 = 0
\]
\[
x = 3
\]

Thus, the function is undefined at \( x = 3 \), meaning the domain excludes \( x = 3 \). In interval notation, the domain is:

\[
(-\infty, 3) \cup (3, \infty)
\]

---

Would you like more details on any of these solutions, or help with another question?

### 5 Relative Questions:
1. How would the domain change if the function were \( f(x) = \sqrt{19 - x} \)?
2. How do we determine the domain of logarithmic functions?
3. How can we check if a function has vertical asymptotes?
4. What happens to the domain if we add a constant to the square root function?
5. How does the equation \( e^{t-2} = 1 \) change if we replace 1 with another constant?

### Tip:
Always check for restrictions like square roots and denominators when finding the domain of a function.

Find the domain of the function f(x) = √(x - 19), solve 5e^(t-2) = 5 using logarithms, and find the domain of the function f(x) = (4x + 6)/(x - 3).

This page provides solutions to domain-related problems for square root and rational functions, as well as step-by-step solutions for solving exponential equations using logarithms. Learn the key concepts and formulas involved in these algebraic problems, suitable for high school students in Grades 10-12.

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