Problem 64 - Algebra and Trigonometry

Question

\(59-64\) Find the domain of the function. $$ h(x)=\sqrt{x-2}-\log _{5}(10-x) $$

Step-by-Step Solution

Answer

The domain of the function is \( 2 \leq x < 10 \).

Step 1: Determine domain of square root

The domain of the square root function \( \sqrt{x-2} \) requires that the expression inside the square root is non-negative. This gives us the inequality:\[ x - 2 \geq 0 \]Solving for \( x \), we find:\[ x \geq 2 \]

Step 2: Determine domain of logarithm

The domain of the logarithmic function \( \log_{5}(10-x) \) requires that the argument is positive. This gives us the inequality:\[ 10 - x > 0 \]Solving for \( x \), we find:\[ x < 10 \]

Step 3: Combine domains

The domain of the function \( h(x) = \sqrt{x-2} - \log_{5}(10-x) \) is the set of values where both domain conditions, \( x \geq 2 \) and \( x < 10 \), are satisfied simultaneously.The combined domain is:\[ x \geq 2 \quad \text{and} \quad x < 10 \]

Step 4: Write final domain

Expressing this interval using inequality notation, the domain of the function \( h(x) \) is:\[ 2 \leq x < 10 \]

Key Concepts

Functions and Their Properties

A function is a rule that assigns each element in a set, called the domain, to a unique element in another set, called the range. Understanding the properties of functions helps you determine how the function behaves and interacts with other mathematical concepts. One critical property of functions is their domain, which tells you which input values are valid for the function.
For different types of functions, the domain can vary due to the inherent operations involved. For instance, you cannot take the square root of a negative number when dealing with real numbers, and you cannot take the logarithm of zero or a negative number.
When given a composite function like the one in this exercise, it's essential to find the domain for each component separately, and then combine those domains to determine the overall domain.

Square Root Function

The square root function is a common mathematical tool symbolized by the square root symbol \( \sqrt{} \). It returns the number which, when squared, equals the input value. However, within real numbers, you cannot take the square root of a negative value, as it does not produce a real number.
For the square root function \( \sqrt{x-2} \), the condition \( x-2 \geq 0 \) needs to be met to ensure the result is a real number. Solving this gives \( x \geq 2 \). Therefore, any value less than 2 would result in attempting to take the square root of a negative number, which is not within the scope of real numbers.
Understanding such properties is crucial when determining the domain for functions involving square roots.

Logarithmic Function

The logarithmic function, like \( \log_{b}(x) \), calculates the power to which the base \( b \) must be raised to obtain the number \( x \). However, the function only accepts positive numbers as inputs, meaning \( x > 0 \).
In the exercise, you encounter \( \log_{5}(10-x) \), requiring \( 10-x > 0 \). Solving this inequality, we find that \( x < 10 \). This condition is necessary because logarithms of zero or negative numbers are undefined in the real number system.
Recognizing the nature of logarithmic functions and their domains is a valuable skill when performing various mathematical operations.