Problem 24 - College Algebra

Question

Given functions \(q(x)=\frac{1}{\sqrt{x}}\) and \(h(x)=x^{2}-9,\) state the domain of each of the following functions using interval notation. a. \(\frac{q(x)}{h(x)}\) b. \(q(h(x))\) c. \(h(q(x))\)

Step-by-Step Solution

Answer

a. \((0, 3) \cup (3, \infty)\); b. \((-\infty, -3) \cup (3, \infty)\); c. \((0, \infty)\)

Step 1: Determine Domain of q(x)

The function \( q(x) = \frac{1}{\sqrt{x}} \) is only defined for \( x > 0 \) because the square root and the denominator cannot be zero. Thus, the domain of \( q(x) \) is \((0, \infty)\).

Step 2: Determine Domain of h(x)

The function \( h(x) = x^2 - 9 \) is a polynomial and is defined for all real numbers. Hence, the domain of \( h(x) \) is \((-\infty, \infty)\).

Step 3: Identify Domain Constraints for q(x)/h(x)

For \( \frac{q(x)}{h(x)} = \frac{1/\sqrt{x}}{x^2 - 9} \), \( q(x) \) requires \( x > 0 \) and \( h(x) eq 0 \). Solve \( x^2 - 9 = 0 \) to find when \( h(x) = 0 \), which gives \( x = \pm 3 \). The domain is \( x > 0 \) excluding \( x = 3 \), so the domain is \((0, 3) \cup (3, \infty)\).

Step 4: Identify Domain Constraints for q(h(x))

For \( q(h(x)) = \frac{1}{\sqrt{h(x)}} = \frac{1}{\sqrt{x^2 - 9}} \), solve \( x^2 - 9 > 0 \) which leads to \( x^2 > 9 \) or \( x > 3 \) and \( x < -3 \). Hence, the domain is \((-\infty, -3) \cup (3, \infty)\).

Step 5: Identify Domain Constraints for h(q(x))

For \( h(q(x)) = \left( \frac{1}{\sqrt{x}} \right)^2 - 9 = \frac{1}{x} - 9 \), it is defined for \( q(x) \) which requires \( x > 0 \). Therefore, the domain is \((0, \infty)\).

Key Concepts

Interval Notation

When expressing the domain of functions, interval notation is a clean and concise way to show which input values are acceptable for a function. It takes the form of either open intervals, closed intervals, or a combination of both.
In interval notation:

• An open interval, such as \(a, b\), means that the endpoints 'a' and 'b' are not included.
• A closed interval, indicated by \[a, b\], includes both endpoints.
• If a function is defined for all real numbers except specific points, we use union symbols \cup to combine different intervals.

For instance, if a function is defined from negative infinity up to but not including -3 and from 3 to infinity, we write the domain as \((-\infty, -3) \cup (3, \infty)\). This helps in clearly conveying the function's domain.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. This means if you have two functions, say \( f(x) \) and \( g(x) \), you can form a composite function \( (f \circ g)(x) \), which is read as 'f composed with g'.
Here's how it generally works:

• The output of \( g(x) \) becomes the input for \( f(x) \).
• The result is a new function that we represent as \( f(g(x)) \).

When dealing with composite functions, determining the domain can be a bit tricky. It's essential first that the function \( g(x) \) is defined for all its input values. Additionally, \( f(x) \) must be defined for the output values of \( g(x) \). Thus, a composite function like \( q(h(x)) \) involves ensuring both conditions are met.

Polynomial Functions

Polynomial functions are expressions that involve terms in which variables are raised to whole number exponents. A typical polynomial looks like \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_i \) are coefficients.
Key characteristics of polynomial functions include:

• They are defined for all real numbers.
• Their graphs are smooth and continuous without breaks, holes, or jumps.
• The domain of a polynomial function is always \((-\infty, \infty)\).

For example, the function \( h(x) = x^2 - 9 \) is a polynomial. Its domain covers all real numbers, but when composing with another function, domain restrictions may appear due to the other function.

Square Root Functions

A square root function takes the form \( \sqrt{x} \), where the result is the non-negative number that, when squared, gives \( x \). The domain of a square root function is typically restricted to non-negative values, i.e., \( x \geq 0 \) because the square root of a negative number is not a real number.
Features of square root functions include:

• The output is always non-negative.
• The graph of a basic square root function \( y = \sqrt{x} \) starts at the origin and increases without bound.

In contexts involving more complex functions like \( q(x) = \frac{1}{\sqrt{x}} \), additional restrictions apply because you also cannot have a zero in the denominator. Therefore, the function \( q(x) \) is defined for \( x > 0 \). This means both the non-negative restriction of the square root and the non-zero restriction of the division operation need to be considered to determine the domain.