Problem 510 - Precalculus

Question

State the domain, vertical asymptote, and end behavior of the function \(g(x)=\ln (4 x+20)-17\).

Step-by-Step Solution

Answer

Domain: \((-5, \infty)\), Vertical Asymptote: \(x = -5\), End Behavior: \(\lim_{x\to\infty} g(x) = \infty\).

Step 1: Find the Domain

The function is defined whenever the argument of the logarithm is positive. For the function \( g(x) = \ln(4x + 20) - 17 \), we need \(4x + 20 > 0\). Solve this inequality:\[4x + 20 > 0 \4x > -20 \x > -5 \\]So, the domain of \( g(x) \) is all real numbers greater than \(-5\), or in interval notation, \((-5, \infty)\).

Step 2: Determine Vertical Asymptote

A vertical asymptote occurs where the logarithmic function is undefined, such as when the argument becomes zero. For \(g(x) = \ln(4x + 20) - 17\), set \(4x + 20 = 0\):\[4x + 20 = 0 \4x = -20 \x = -5 \\]Thus, \(x = -5\) is the vertical asymptote.

Step 3: Analyze End Behavior

The end behavior of the function describes what happens to \(g(x)\) as \(x\) approaches positive infinity. As \(x\to \infty\), the term \(4x + 20\to \infty\), so the logarithm, \(\ln(4x + 20)\to \infty\). Thus, the function \(g(x) = \ln(4x + 20) - 17\) approaches infinity, as well:\[\lim_{{x\to\infty}} g(x) = \infty\]

Key Concepts

Domain of a Function

The domain of a function simply refers to all the possible values that the input, or "x-values", can take. For logarithmic functions, it's crucial to ensure the argument of the logarithm is positive, because the logarithm of 0 or a negative number is undefined.
For the function given, \( g(x) = \ln(4x + 20) - 17 \), we need the input of the logarithm, or \(4x + 20\), to be greater than 0. Solving the inequality \(4x + 20 > 0\) helps us find this range of values.
Here are the steps:

• Start with \(4x + 20 > 0\).
• Subtract 20 from both sides to get \(4x > -20\).
• Divide both sides by 4, yielding \(x > -5\).

So, the domain of the function \(g(x)\) is all real numbers greater than \(-5\), or in interval notation, \((-5, \infty)\). This means the function is defined for every point in this range, ensuring the logarithm has a positive argument.

Vertical Asymptotes

Vertical asymptotes are locations where a function's value heads towards infinity, either positively or negatively, and they indicate a point where the function is undefined. For logarithmic functions like \( g(x) = \ln(4x + 20) - 17 \), this typically occurs when the argument of the logarithm reaches zero.
To find the vertical asymptote, set the argument equal to zero: \(4x + 20 = 0\). Solving this:

• Move 20 to the other side: \(4x = -20\).
• Divide by 4: \(x = -5\).

Thus, \(x = -5\) is the vertical asymptote for the function \(g(x)\). This means the function approaches either positive or negative infinity as \(x\) gets closer to \(-5\), but never actually reaches a definable value at \(-5\). It marks a distinct separation in the function's graph.

End Behavior of Functions

The end behavior of a function describes the trend of the function's output values, \(g(x)\), as the input \(x\) moves towards infinity or negative infinity. It's a way to predict how the function behaves over very large or very small inputs.
For the function \(g(x) = \ln(4x + 20) - 17\), we focus on what happens as \(x\to \infty\):

• The expression \(4x + 20\) grows indefinitely large.
• Thus, \(\ln(4x + 20)\) also heads towards infinity because the logarithm of a very large number is itself large.

Since \(g(x) = \ln(4x + 20) - 17\), subtracting 17 shifts everything downwards by 17 units, but it still climbs towards infinity eventually. Therefore, the function's end behavior as \(x\to \infty\) is that \(g(x)\to \infty\).
End behavior helps us understand the function's broader picture, knowing where it trends as inputs grow large or very small.