Question: Exercise
g(x) = -6 / (x^2 - 5)^2
g(x) = -3 \sqrt{x^4 - z}
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1. **State the problem:** We are given two functions:
$$g(x) = \frac{-6}{(x^2 - 5)^2}$$
and
$$g(x) = -3 \sqrt{x^4 - z}$$
We want to analyze or simplify these expressions and understand their forms.
2. **Analyze the first function:**
The first function is a rational function with a squared denominator:
$$g(x) = \frac{-6}{(x^2 - 5)^2}$$
This means the denominator is the square of the binomial $x^2 - 5$.
3. **Analyze the second function:**
The second function involves a square root:
$$g(x) = -3 \sqrt{x^4 - z}$$
Here, $z$ is a constant or variable, and the expression inside the root is $x^4 - z$.
4. **Important notes:**
- For the first function, the denominator cannot be zero, so $x^2 - 5 \neq 0 \Rightarrow x \neq \pm \sqrt{5}$.
- For the second function, the expression inside the square root must be non-negative for real values:
$$x^4 - z \geq 0 \Rightarrow x^4 \geq z$$
5. **No further simplification is possible without additional information about $z$.**
6. **Summary:**
- The first function is a rational function with a squared quadratic denominator.
- The second function is a radical function involving the fourth power of $x$ minus $z$.
- Domain restrictions apply as explained.
**Final answer:**
$$g(x) = \frac{-6}{(x^2 - 5)^2}$$
and
$$g(x) = -3 \sqrt{x^4 - z}$$