Problem 197 - Precalculus

Question

For the following exercises, state the domain and range of the function. $$h(x)=\ln (4 x+17)-5$$

Step-by-Step Solution

Answer

Domain: \(x > -\frac{17}{4}\); Range: all real numbers.

Step 1: Identify the Function Type

The given function is \( h(x) = \ln(4x + 17) - 5 \). This is a logarithmic function, specifically the natural logarithm.

Step 2: Determine the Domain

For a function \( \ln(y) \) to be defined, \( y \) must be greater than zero. Thus, we set up the inequality:\[4x + 17 > 0\]Solving this inequality, we subtract 17 from both sides:\[4x > -17\]Then, divide by 4:\[x > -\frac{17}{4}\]So, the domain of the function is \( x > -\frac{17}{4} \).

Step 3: Determine the Range

The natural logarithm function \( \ln(y) \) can output any real number when \( y > 0 \). Subtracting 5 from any real number still results in a real number. Therefore, the range of \( h(x) \) is all real numbers or \( (-\infty, \infty) \).

Key Concepts

Domain of a Function

The domain of a function refers to all the possible input values, typically represented by the variable \( x \), that make a function meaningful or defined. When dealing with logarithmic functions, like the natural logarithm \( \ln(y) \), the primary concern is ensuring the argument inside the logarithm is greater than zero.
For example, for the function \( h(x) = \ln(4x + 17) - 5 \), inside the logarithm \( 4x + 17 \) must be positive. We express this with the inequality:

• \( 4x + 17 > 0 \)

This inequality must be solved to find the domain of the function.
Simplify it step by step:

• Subtract 17 from both sides: \( 4x > -17 \).
• Divide by 4: \( x > -\frac{17}{4} \).

Thus, the domain is all \( x \) values greater than \( -\frac{17}{4} \), which means the function can be evaluated for any value greater than \( -4.25 \).

Range of a Function

The range of a function is the set of possible output values that result from using every domain value in a function. For a natural logarithmic function like \( \ln(y) \), the range comprises all real numbers, since \( \ln(y) \) can produce any real number as long as \( y > 0 \).
In our specific function \( h(x) = \ln(4x + 17) - 5 \):

• The expression \( \ln(4x + 17) \) covers all real numbers from \(-\infty\) to \(\infty\) when \( 4x + 17 > 0 \).
• Subtracting 5 from these real numbers (which represents a vertical shift downwards in the graph of the function) also results in a set of all real numbers.

Therefore, the range of \( h(x) \) is \((-\infty, \infty)\), meaning the function's outputs span from negative infinity to positive infinity.

Natural Logarithm

The natural logarithm is a particular type of logarithm that uses Euler's number \( e \) (approximately 2.718) as its base. It's denoted as \( \ln(x) \).

• The natural logarithm is particularly important in calculus and mathematical modeling, as it appears naturally in many growth and decay processes.
• It answers the question: "To what power must \( e \) be raised to produce a given number?"

For example, the equation \( \ln(e) = 1 \) because \( e^1 = e \).
In the given function \( h(x) = \ln(4x + 17) - 5 \), the logarithm's role is to take \( 4x + 17 \) and transform it to an exponent of \( e \).
The logarithm helps translate multiplication into addition, division into subtraction, and powers to products, making complex calculations more manageable.
Understanding the properties of the natural logarithm can help unravel and solve various mathematical problems efficiently.