Question

for the given function, state if its increasing (i) or decreasing (d).
as well as the correct domain and range:

• all real numbers : r
• greater/less than or equal to :

$x>=h$ ; $y>=k$ ; $x<=h$ ; $y<=k$
$f(x)=2(x+4)^3+5$
increasing or decreasing:
domain:
range:

Explanation:

Step1: Analyze monotonicity via derivative

Find derivative: $f'(x) = 2 \cdot 3(x+4)^2 = 6(x+4)^2$

Step2: Evaluate derivative sign

$6(x+4)^2 \geq 0$ for all $x$, equals 0 only at $x=-4$, so $f(x)$ is increasing everywhere.

Step3: Determine domain

No restrictions on $x$ for cubic functions, so domain is all real numbers $\mathbb{R}$.

Step4: Determine range

Cubic functions map all real $x$ to all real $y$, so range is all real numbers $\mathbb{R}$.

Answer:

Increasing or Decreasing: I
Domain: $\mathbb{R}$
Range: $\mathbb{R}$