Functions with Roots

Determine the domain of definition of the following function with an even-index root:

$f(x) = \sqrt{3x - 12}$

Find the domain of the quadratic function under a square root:

$h(x) = \sqrt{16 - x^2}$

Calculate the domain of the function containing an even-index root in the denominator:

$f(x) = \frac{1}{\sqrt[4]{x^2 - 5x + 6}}$

Calculate the domain of definition of the function with an odd-index root:

$g(x) = \sqrt[3]{2x^2 - 8x + 1}$

Find the domain of the irrational function containing a rational expression:

$f(x) = \sqrt{\frac{x - 2}{x + 5}}$

Find the domain of definition of the function formed by the sum of two roots:

$f(x) = \sqrt{x + 4} + \sqrt{6 - 2x}$

Determine the domain of the irrational function whose radicand is an algebraic fraction with a quadratic numerator:

$f(x) = \sqrt{\frac{x^2 - 9}{x - 1}}$

Find the domain of definition of the nested function:

$f(x) = \sqrt{2 - \sqrt{x - 3}}$

Determine the domain of the irrational function containing a rational expression where the numerator and denominator share roots:

$f(x) = \sqrt{\frac{x^3 - 3x^2 - 4x + 12}{x^2 + x - 2}}$

Solve the inequality associated with the domain of the function $f(x) = \sqrt{\sqrt{x + 7} - x - 1}$ and express it in interval notation.

Determine the values of the parameter $k$ for which the domain of the function $f(x) = \sqrt{x^2 + kx + 9}$ is the entire set of real numbers $\mathbb{R}$.

From the function $f(x) = \sqrt{x}$, write the algebraic expression for the function $g(x)$ whose graph is obtained by translating $f(x)$ three units to the left and two units upwards. What is the domain of $g(x)$?

Determine the domain and range of the function with a negative leading coefficient:

$f(x) = -\sqrt{x - 1}$

Observe the attached graph representing a function of the form $f(x) = a\sqrt{x - b} + c$. Determine the values of the parameters $a$, $b$ and $c$, knowing that the vertex of the semi-parabola is located at the point $(2, 1)$ and it passes through the point $(6, -3)$.

Find the axis intercepts of the function:

$f(x) = \sqrt{x + 9} - 3$

Calculate the points of intersection between the irrational function $f(x) = \sqrt{2x + 5}$ and the linear function $g(x) = x + 1$.

Given the function $f(x) = 3 + \sqrt{2x - 4}$, determine its inverse function $f^{-1}(x)$ and state the domain and range of this inverse function.

Given the function $f(x) = \sqrt{x^2 - 4x}$, calculate its inverse function $f^{-1}(x)$ by restricting the original domain to the interval $(-\infty, 0]$. Clearly state the domain and range of the resulting inverse function.

The relationship between the area $A$ of a circle and its radius $r$ is given by the formula $A = \pi r^2$. Express the radius in terms of the area, $r(A)$, and state the valid domain of definition in this geometric context.

The period $T$ (in seconds) of a simple pendulum can be approximated by the function $T(L) = 2\pi\sqrt{\frac{L}{9.8}}$, where $L$ is the length of the pendulum in meters. Calculate the length required for the period to be exactly $2$ seconds and state the theoretical domain of the function $T(L)$ in this physical context.

The production cost $C(x)$, in pesos, of $x$ parts of an electronic component for satellites follows a model of diminishing returns given by $C(x) = 50\,000 + 2\,400\sqrt{x - 50}$. Determine the valid domain for the cost function and calculate how many parts were manufactured if the total cost amounted to $98\,000$ pesos.

In the theory of special relativity, the Lorentz factor $\gamma$ depends on the velocity $v$ of an object according to the function $\gamma(v) = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $c$ is the speed of light in a vacuum (approximately $3 \times 10^8$ m/s). Determine the domain of the function $\gamma(v)$ and interpret physically what happens to the function as $v$ approaches the boundaries of this domain.

A rectangle is inscribed in the region bounded by the x-axis and the upper semicircle of radius $R = 5$, whose equation is $y = \sqrt{25 - x^2}$. Express the perimetre of the rectangle $P(x)$ in terms of the coordinate $x$ of the vertex located in the first quadrant. Determine the valid domain of $P(x)$ in this geometric context and calculate its value for $x = 3$.

An engineering company estimates that the daily revenue $I(x)$ and costs $C(x)$, in thousands of pesos, when manufacturing $x$ metric tons of a composite material are given by $I(x) = x\sqrt{40 - x}$ and $C(x) = 2x + 5$. Determine the domain of the profit function $B(x) = I(x) - C(x)$ in the context of the problem and calculate whether the company makes a profit or a loss when manufacturing $15$ metric tons.