Problem 21 - Calculus Volume 1

Question

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. $$ g(x)=\sqrt{\frac{7}{x-5}} $$

Step-by-Step Solution

Answer

Domain: \( x > 5 \), Range: \((0, \infty)\), No zeros/intercepts.

Step 1: Determine the Domain

The function is defined wherever the expression under the square root is non-negative and the denominator is not zero. Solving for the domain, set \( \frac{7}{x-5} \geq 0 \). The fraction \( \frac{7}{x-5} \) is positive when \( x > 5 \) and undefined at \( x = 5 \). Thus, the domain of \( g(x) \) is \( x > 5 \).

Step 2: Determine the Range

Since \( g(x) = \sqrt{\frac{7}{x-5}} \), we see that \( \sqrt{\frac{7}{x-5}} \geq 0 \). As \( x \to \infty \), \( \frac{7}{x-5} \to 0 \) leading \( g(x) \to 0 \). Since \( x > 5 \), \( g(x) \) can take any positive real number, so the range is \( g(x) > 0 \). Therefore, the range is \((0, \infty)\).

Step 3: Find the Zeros/Intercepts

The zeros of the function occur where \( g(x) = 0 \). Since \( \sqrt{\frac{7}{x-5}} \) is always positive for \( x > 5 \), the function never reaches zero. Therefore, there are no zeros or x-intercepts. Similarly, since the function is not defined at \( x=5 \), there is no y-intercept as the domain does not include \( x=0 \).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values (usually represented by \( x \)) for which the function is defined. When dealing with functions involving fractions and square roots, determining the domain requires a bit more attention.
In the case of the function \( g(x) = \sqrt{\frac{7}{x-5}} \), the function is affected by two conditions:

• The denominator \( x - 5 \) cannot be zero, as division by zero is undefined. Thus, \( x eq 5 \).
• The expression under the square root, \( \frac{7}{x-5} \), must be non-negative because the square root of a negative number is not defined in the set of real numbers.

By solving \( \frac{7}{x-5} \geq 0 \), we find that \( x \) must be greater than 5.
Therefore, the domain of this function is \( x > 5 \). This ensures the value under the square root is positive, and the denominator is non-zero.

Range of a Function

The range of a function consists of all possible output values (often represented by \( y \)) that the function can produce. When assessing range, especially in functions involving square roots, it's important to consider how the function behaves as \( x \) changes.
For the function \( g(x) = \sqrt{\frac{7}{x-5}} \), observe that:

• The expression \( \sqrt{\frac{7}{x-5}} \) is non-negative because square roots yield non-negative results for non-negative inputs.
• As \( x \to \infty \), \( \frac{7}{x-5} \to 0 \), causing \( g(x) \to 0 \).
• Since the domain restricts \( x > 5 \), \( g(x) \) will never reach 0; it starts approaching zero from any positive value as \( x \) increases.

The range of \( g(x) \) is all positive real numbers, represented as \((0, \infty)\). This means the function produces any positive number as an output but never zero or negative results.

Zeros of a Function

The zeros, or x-intercepts, of a function are the points where the function equals zero, i.e., where \( g(x) = 0 \). These values of \( x \) make the output zero. In some cases, functions may not have zeros.
Examining the function \( g(x) = \sqrt{\frac{7}{x-5}} \), we need to see if there's any \( x \) that makes the output zero:

• Due to the square root form, \( g(x) \) is positive whenever it's defined (based on the domain \( x > 5 \)).
• This positivity ensures that \( g(x) \) cannot equal zero within its domain.

Consequently, the function \( g(x) \) has no zeros, as it always yields positive values for inputs within its defined range.
Moreover, since the function does not include \( x = 5 \) in its domain, it doesn’t cross through the x-axis at all, meaning there are also no y-intercepts related to typical functions.