Chapter 10.2, Problem 57PPSE

Solution

(a.)

What part of a parabola is modeled by the given function.

It has been determined that the given function models the upward branch of the parabola $y^2=x$ .

Given:

The function, $y=\sqrt{x}$ .

Concept used:

A parabola has two parts (branches) divided by the vertex; namely the left and right branches for a vertical parabola and the upward and downward branch for a horizontal parabola.

Calculation:

The given function is $y=\sqrt{x}$ .

Squaring both sides,

  $y^2=x$

This is the original parabola. Specifically, this is a horizontal parabola which opens rightward.

The vertex of this parabola is at the origin, which divides the parabola into two parts; the upward part given by $y=\sqrt{x}$ and the downward part given by $y=-\sqrt{x}$ .

So, the given function models the upward branch of the parabola $y^2=x$ .

Conclusion:

It has been determined that the given function models the upward branch of the parabola $y^2=x$ .

(b.)

The domain and range for the given function.

It has been determined that both domain and range of the given function is $\left[0,\infty \right)$ .

Given:

The function, $y=\sqrt{x}$ .

Concept used:

The quantity under the square root is always non-negative if the resulting value is real.

Calculation:

The given function is $y=\sqrt{x}$ .

As discussed, the quantity under the square root is always non-negative if the resulting value is real.

This implies that for $\sqrt{x}$ to be defined and real, it must follow that $x\ge 0$ .

So, the domain of the function is $\left[0,\infty \right)$ .

Now, $y=\sqrt{x}$ implies that $y$ represents only the positive square root of $x$ , which further implies that $y\ge 0$ .

So, the range of the function is also $\left[0,\infty \right)$ .

Conclusion:

It has been determined that both domain and range of the given function is $\left[0,\infty \right)$ .