Problem 9 - College Algebra

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: \((-\infty, \infty)\).

Step 1: Understand the function

The given function is a natural logarithm function, shifted vertically. We will analyze how the expression inside the logarithm impacts the domain, and how the constants affect the range.

Step 2: Find the domain

The natural logarithm function, represented by \((x)\), is only defined for positive values. Hence, we consider the inequality \(4x + 17 > 0\) to determine the domain. Solving this inequality gives us \(x > -\frac{17}{4}\). Thus, the domain of \(h(x)\) is \(x > -\frac{17}{4}\) or in interval notation \(\left(-\frac{17}{4}, \infty\right)\).

Step 3: Determine the range

The natural logarithm function, \((x)\), has a range of all real numbers \((-\infty, \infty)\). The transformation \(\ln(4x+17) -5\) shifts the entire function downwards by 5 units, however, it does not compress the range. Therefore, the range of \(h(x)\) remains \((-\infty, \infty)\).

Key Concepts

Natural Logarithm Function

The natural logarithm function, represented by \(\ln(x)\), is a logarithm with a base of \(e\), where \(e \approx 2.718\). It is a fundamental function in mathematics and is used extensively in calculus and algebra. The natural logarithm is only defined for positive real numbers, which is an important property when determining its domain. For the general form \(\ln(g(x))\), \(g(x)\) must be greater than zero.In the exercise, the function given is \(h(x) = \ln(4x+17) - 5\). Let's break it down:- Inside the logarithm, we have \(4x + 17\). This expression must be positive, which means we set up the inequality \(4x + 17 > 0\) to find valid \(x\) values.- The subtraction of 5 is a vertical shift and does not affect the domain, just alters the vertical positioning of the function.

Function Transformation

Function transformations are alterations that change the appearance of the graph of a function. Common transformations include shifts, reflections, stretches, and compressions. Understanding these helps analyze how changes in a function's formula affect its graph.For the function \(h(x) = \ln(4x+17) - 5\):- **Vertical Shift**: The \(-5\) at the end signifies a downward shift. Each point on the graph of \(\ln(4x+17)\) moves down 5 units, altering its vertical positioning without affecting the domain.- **Domain Consideration**: Since \(\ln(x)\) is only defined for positive arguments, the expression \(4x + 17 > 0\) is critical to figuring out where the curve exists on the \(x\)-axis. Solving it, we get \(x > -\frac{17}{4}\), confirmed as the start of our domain.

Interval Notation

Interval notation is a method of writing down a set of numbers, specifically those that form an interval on the number line. It addresses range and domain in a concise way, employing brackets and parentheses.- A **parenthesis \(()\)** indicates that the endpoint is not included in the interval, while a **bracket \([]\)** means it is included. For example, \(x > a\) would be represented as \((a, \infty)\).For the exercise, the domain of \(h(x) = \ln(4x+17) - 5\) is expressed using interval notation as \((-\frac{17}{4}, \infty)\). This signifies that \(x\) starts just beyond \(-\frac{17}{4}\) and extends infinitely to the right. The range, not constrained by the transformation except for a shift, remains \((-\infty, \infty)\) just like the basic \(\ln(x)\) function.