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 :

$x>=h ; y>=k ; x<=h ; y<=k$
$f(x) = -7\sqrt3{x-3}+5$
increasing or decreasing:
domain:
range:

Explanation:

Step1: Analyze monotonicity

The parent function $\sqrt[3]{x}$ is increasing. The given function has a negative coefficient $-7$, so it reflects the parent function over the x-axis, reversing the monotonicity. Thus, $f(x)$ is decreasing.

Step2: Find domain of $f(x)$

Cube roots are defined for all real numbers. The expression $x-3$ is a real number for all $x\in\mathbb{R}$. So domain is all real numbers: $\mathbb{R}$.

Step3: Find range of $f(x)$

The range of $\sqrt[3]{x-3}$ is $\mathbb{R}$. Multiplying by $-7$ gives all real numbers, then adding $5$ still results in all real numbers. So range is $\mathbb{R}$.

Answer:

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