https://file.aiquickdraw.com/mathbot/file/17275227629998nyw81ld.jpg

Let's analyze the functions given in the image and find their domains of definition.

### (i) \( g(x) = \frac{x^3}{x^2 - 3x + 2} \)

This is a rational function. The domain is all real values of \( x \) except where the denominator becomes zero. To find the values that make the denominator zero, we need to solve the equation:

\[
x^2 - 3x + 2 = 0
\]

Factoring the quadratic:

\[
x^2 - 3x + 2 = (x - 1)(x - 2) = 0
\]

So, \( x = 1 \) and \( x = 2 \) make the denominator zero. Therefore, the domain of \( g(x) \) is:

\[
x \in (-\infty, 1) \cup (1, 2) \cup (2, \infty)
\]

### (ii) \( F(t) = \sqrt{2 - \sqrt{t}} \)

For the function \( F(t) \), we must ensure that the expressions inside both square roots are non-negative.

1. The inner square root, \( \sqrt{t} \), requires that \( t \geq 0 \).
2. The expression inside the outer square root, \( 2 - \sqrt{t} \), must also be non-negative:
   \[
   2 - \sqrt{t} \geq 0
   \]
   Solving this inequality:
   \[
   \sqrt{t} \leq 2
   \]
   Squaring both sides:
   \[
   t \leq 4
   \]

Combining both conditions, we get \( 0 \leq t \leq 4 \).

Thus, the domain of \( F(t) \) is:
\[
t \in [0, 4]
\]

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Would you like any further clarification or more details?

### Related Questions:
1. How do you solve more complicated quadratic equations when finding domains of rational functions?
2. What are the implications of having complex numbers in the domain of a square root function?
3. How would the domain of \( g(x) \) change if the denominator were a cubic polynomial instead of quadratic?
4. Can you explain how transformations affect the domain of a square root function?
5. How do you deal with domains for functions involving higher-degree radicals (e.g., cube roots)?

### Tip:
When dealing with rational functions, always check the denominator for values that could cause division by zero, as those will not be in the domain.

Find the domain of the functions: (i) g(x) = x^3 / (x^2 - 3x + 2), (ii) F(t) = sqrt(2 - sqrt(t))

Learn how to find the domain of two functions: a rational function g(x) = x^3 / (x^2 - 3x + 2) and a radical function F(t) = sqrt(2 - sqrt(t)). This problem covers factoring quadratics and ensuring the non-negativity of square roots. Suitable for Grades 9-12.

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