Question

the graph of $f(x) = \sqrt3{x + 8}$ is shown.
which statement is true?
\bigcirc the function is only increasing when $x \geq -8$.
\bigcirc the function is only increasing when $x \geq 0$.
\bigcirc the function is always decreasing.
\bigcirc the function is always increasing.

Explanation:

Step1: Analyze the function type

The function \( f(x)=\sqrt[3]{x + 8}\) is a cube - root function. The general form of a cube - root function is \(y = \sqrt[3]{x - h}+k\), which is a transformation of the parent function \(y=\sqrt[3]{x}\). The parent function \(y = \sqrt[3]{x}\) has a domain of all real numbers and is always increasing. For the function \(y=\sqrt[3]{x - h}+k\), the horizontal shift (by \(h\) units) and vertical shift (by \(k\) units) do not change the increasing/decreasing nature of the cube - root function. In our case, \(h=- 8\) and \(k = 0\), so \(f(x)=\sqrt[3]{x+8}=\sqrt[3]{x-(-8)}\).

Step2: Analyze the graph

Looking at the given graph of \(f(x)=\sqrt[3]{x + 8}\), as we move from left to right (as \(x\) increases), the \(y\) - values (the function values) are always increasing. Let's check the options:

• Option 1: "The function is only increasing when \(x\geq - 8\)" is wrong. Because if we take \(x\lt - 8\), say \(x=-16\), \(f(-16)=\sqrt[3]{-16 + 8}=\sqrt[3]{-8}=-2\), and if we take \(x=-12\), \(f(-12)=\sqrt[3]{-12 + 8}=\sqrt[3]{-4}\approx - 1.59\). Since \(-1.59\gt - 2\), when \(x\) increases from \(-16\) to \(-12\) (where \(x\lt - 8\)), the function is increasing.
• Option 2: "The function is only increasing when \(x\geq0\)" is wrong. For \(x=-4\), \(f(-4)=\sqrt[3]{-4 + 8}=\sqrt[3]{4}\approx1.59\), and for \(x = 0\), \(f(0)=\sqrt[3]{0 + 8}=2\). Also, for \(x\lt0\), say \(x=-8\), \(f(-8)=\sqrt[3]{-8 + 8}=0\), and for \(x=-4\) (which is greater than \(-8\) but less than \(0\)), \(f(-4)\approx1.59\gt0\), so the function is increasing for \(x\lt0\) as well.
• Option 3: "The function is always decreasing" is wrong. Since as \(x\) increases, \(f(x)\) increases, not decreases.
• Option 4: "The function is always increasing" is correct, as we can see from the analysis of the function type (cube - root functions are always increasing) and from the graph (as \(x\) increases, \(y\) increases for all real \(x\)).

Answer:

The function is always increasing.