Question

graph the following function. state the domain and range. is the function increasing or decreasing?
( h(x) = sqrt3{x} - 4 )

choose the correct graph below

a.

b.

c.

d.

Explanation:

Step1: Analyze the parent function

The parent function of \( h(x)=\sqrt[3]{x}-4 \) is \( y = \sqrt[3]{x} \), which has a domain of all real numbers (\( (-\infty, \infty) \)) and a range of all real numbers (\( (-\infty, \infty) \)). The graph of \( y=\sqrt[3]{x} \) passes through the origin \((0,0)\), is increasing, and has a point - symmetric shape about the origin.

Step2: Analyze the transformation

For the function \( h(x)=\sqrt[3]{x}-4 \), this is a vertical shift of the parent function \( y = \sqrt[3]{x} \) down by 4 units. So, when \( x = 0 \), \( h(0)=\sqrt[3]{0}-4=- 4 \). The domain of \( h(x) \) is still all real numbers (\( (-\infty,\infty) \)) because we can take the cube - root of any real number, and the range is also all real numbers (\( (-\infty,\infty) \)) since the vertical shift does not restrict the output values. The function is increasing because the parent function \( y = \sqrt[3]{x} \) is increasing and a vertical shift does not change the increasing/decreasing nature of a function.

Step3: Analyze the graphs

• Graph A: The shape does not match the shape of a cube - root function. A cube - root function has a smooth, S - like (or point - symmetric) shape, while this graph looks more like a piece - wise function with a different behavior.
• Graph B: The horizontal shift or other transformations do not match. The key point at \( x = 0 \) should be \( y=-4 \), but this graph does not seem to have that property.
• Graph C: The key point at \( x = 0 \) is not \( y = - 4 \), so it does not match the function \( h(x)=\sqrt[3]{x}-4 \).
• Graph D: The graph has the shape of a cube - root function (smooth, increasing) and when \( x = 0 \), \( y=-4 \), which matches the function \( h(x)=\sqrt[3]{x}-4 \).

Answer:

Domain:

The domain of \( h(x)=\sqrt[3]{x}-4 \) is all real numbers. In interval notation, the domain is \( (-\infty,\infty) \).

Range:

The range of \( h(x)=\sqrt[3]{x}-4 \) is all real numbers. In interval notation, the range is \( (-\infty,\infty) \).

Increasing/Decreasing:

The function \( h(x)=\sqrt[3]{x}-4 \) is increasing.

Correct Graph:

D