Question

Find the natural domain for each of the following. (a) \(F(z)=\sqrt{2 z+3}\) (b) \(g(v)=1 /(4 v-1)\) (c) \(\psi(x)=\sqrt{x^{2}-9}\) (d) \(H(y)=-\sqrt{625-y^{4}}\)

Short answer

(a) [-3/2, ∞), (b) R \ {1/4}, (c) (-∞, -3] ∪ [3, ∞), (d) [-5, 5].

Step-by-step solution

Understand Natural Domain

The natural domain of a function is the set of all possible input values (real numbers) for which the function is defined and real.

Solve for Part (a)

For the function \( F(z) = \sqrt{2z + 3} \), the expression under the square root, \( 2z + 3 \), must be non-negative. Thus, we solve the inequality:\[ 2z + 3 \geq 0 \]Subtract 3 from both sides:\[ 2z \geq -3 \]Divide by 2:\[ z \geq -\frac{3}{2} \]Thus, the domain of \( F(z) \) is \( z \geq -\frac{3}{2} \), or \([-\frac{3}{2}, \infty)\).

Solve for Part (b)

For \( g(v) = \frac{1}{4v - 1} \), the denominator \( 4v - 1 \) cannot be zero. Solve the equation:\[ 4v - 1 = 0 \]Add 1 to both sides:\[ 4v = 1 \]Divide by 4:\[ v = \frac{1}{4} \]Therefore, the domain of \( g(v) \) is all real numbers except \( \frac{1}{4} \), or \( (-\infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty) \).

Solve for Part (c)

For \( \psi(x) = \sqrt{x^2 - 9} \), set \( x^2 - 9 \geq 0 \).Add 9 to both sides:\[ x^2 \geq 9 \]This implies \( x \geq 3 \) or \( x \leq -3 \). Therefore, the domain of \( \psi(x) \) is \( (-\infty, -3] \cup [3, \infty) \).

Solve for Part (d)

For \( H(y) = -\sqrt{625 - y^4} \), we need \( 625 - y^4 \geq 0 \).This leads to \( y^4 \leq 625 \).Taking the fourth root of both sides:\[ |y| \leq 5 \]This means \( -5 \leq y \leq 5 \). Therefore, the domain of \( H(y) \) is \([-5, 5]\).

Key concepts

Square Roots

Square roots are fundamental in mathematics, especially when determining the domain of functions involving them. A square root, denoted as \( \sqrt{x} \), represents a number that, when multiplied by itself, gives \( x \). For square roots to give real results, the number under the square root (the radicand) must be non-negative. This stems from the fact that negative numbers don't have real square roots in the set of real numbers.

For example, when determining the domain of a function like \( F(z) = \sqrt{2z + 3} \), we need to ensure that \( 2z + 3 \geq 0 \) because the square root of a negative number is not defined in real numbers. By solving this inequality, we find the values of \( z \) that keep the expression under the root non-negative.

In summary, whenever dealing with square roots, look at the radicand and set it \( \geq 0 \) to find the domain. This process ensures that your function will always return real numbers.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In simpler terms, it's a fraction where the numerator and the denominator are both polynomials.

An essential point when dealing with rational functions is to ensure that the denominator is not zero. This is crucial because division by zero is undefined. Let's consider \( g(v) = \frac{1}{4v - 1} \). To find where this function is defined, solve for \( 4v - 1 = 0 \), since it indicates an undefined point. This gives \( v = \frac{1}{4} \).

• The domain of a rational function is all real numbers except where the denominator equals zero.
• In our example, the domain is \( (-\infty, \frac{1}{4}) \cup (\frac{1}{4}, \infty) \).

By removing these critical values from the domain, the function remains defined and provides real number outputs.

Inequalities

Inequalities are mathematical expressions involving the symbols \(<, \leq, >, \geq\). They describe a range of values rather than specific numbers, and they are pivotal in finding the domain of functions, especially with square roots and rational functions.

For instance, to solve \( \sqrt{x^2 - 9} \), the inequality \( x^2 - 9 \geq 0 \) is used to ensure the expression under the square root remains non-negative. Solving this inequality involves factoring see that \( x \geq 3 \) or \( x \leq -3 \). Hence, the domain is \( (-\infty, -3] \cup [3, \infty) \).

When working with inequalities:

• Identify the expression that needs to be non-zero or non-negative.
• Find the values for which the inequality holds true.

These steps allow you to determine the function's domain accurately.

Real Numbers

Real numbers refer to all the numbers on the number line, including whole numbers, integers, fractions, and irrational numbers like \( \sqrt{2} \). They constitute the most widely used number set in math and are crucial for determining function domains.

When assessing a function's domain, we're primarily concerned with identifying the real number inputs that ensure the function remains defined. For square root and rational functions, this often means excluding numbers that lead to negative square roots or division by zero.

For example, the function \( H(y) = -\sqrt{625 - y^4} \) involves real numbers such as \( y \) values that satisfy \( -5 \leq y \leq 5 \). This range ensures the expression inside the square root is non-negative, keeping it real.

Understanding real numbers and their properties helps bridge the gap between abstract math concepts and practical computations, aiding in accurate function analysis.