Question

Will give brainliest to the correct answer, complete with an in-depth explanation. Thank you!

Answer 1

Answer:

$x \in [1, 6)$

Step-by-step explanation:

Because the function is decreasing:

$g(x_1) > g(x_2) \iff x_1 < x_2$

So:

$g(x^2) > g(5x + 6) \iff x^2 < 5x + 6\\x^2 < 5x + 6\\x^2 - 5x - 6 < 0\\x^2 - 5x + 6.25 - 12.25 < 0\\x^2 - 2\cdot 2.5x + 2.5^2 < 12.25\\(x - 2.5)^2 < 12.25\\x - 2.5 \in (-3.5; 3.5)\\x \in (-1; 6)$

Combining that with the domain $D_g = [1; \infty )$

$x \in [1, 6)$