Question

function analysis > comparing shapes of functions
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compare the following options and select the one that matches the graph
option #1: the graph of the function matches $f(x)=\sqrt3{x}$.
option #2: the graph of the function matches $f(x)=\sqrt{x}$
(1 point)

Explanation:

1. Analyze the domain of the graph: It includes all real numbers (positive, negative, 0), as the curve extends left into negative x-values and right into positive x-values.
2. Check the domain of Option #2: $f(x)=\sqrt{x}$ only has a domain of $x\geq0$, so it cannot match the graph which includes negative x-values.
3. Check the domain and shape of Option #1: $f(x)=\sqrt[3]{x}$ has a domain of all real numbers, and its graph passes through the origin, increasing for all x, with the characteristic shape shown (curving gently, defined for both positive and negative inputs). This matches the given graph.

Answer:

Option #1. The graph of the function matches $f(x) = \sqrt[3]{x}$.