f(x)=ln(x+4), domain, range, graph, and its inverse

TL;DR

The video explains how to find the domain, graph, range, and inverse of the function f(x) = Ln(x) + 4.

Transcript

for this question we are given the function f of X is equal to Ln of X plus 4 and then we have to do these questions I know it has a lot of questions but it's ok because I know you can handle it for the first part is that we are going to find the domain of F this is Ln of X plus 4 Ln is log base e so we are talking about log function so that means ... Read More

Key Insights

• ☺️ The domain of f(x) is given by x > -4.
• ☺️ The graph of the function has a vertical asymptote at x = -4 and approaches positive infinity as x increases.
• ♾️ The range of the function is (-infinity, infinity), indicating that the y-values can be any real number.
• 😀 The inverse of f(x) is given by y = e^x - 4.
• 🧡 The domain of the inverse is the range of the original function.
• 🤪 The graph of the inverse has a horizontal asymptote at y = -4 and goes up towards positive infinity as x increases.
• ♾️ The range of the inverse is (-infinity, infinity), indicating that the y-values can be any real number.

Questions & Answers

Q: How do you determine the domain of f(x) = Ln(x) + 4?

To find the domain, we set the inside of the logarithm function greater than 0, resulting in x > -4. So, the domain of f(x) is (-4, infinity).

Q: What happens when the inside of the logarithm function is 0?

When the inside of the logarithm function is 0, such as in the case of Ln(0), the value is undefined. However, it indicates a vertical asymptote in the graph.

Q: How do you graph the function f(x) = Ln(x) + 4?

By selecting a few x-values and solving for the corresponding y-values, we can plot points on the graph. Connecting these points forms a curve, which approaches a vertical asymptote as x approaches -4.

Q: What is the range of the function f(x) = Ln(x) + 4?

The range of the function is from negative infinity to positive infinity, denoted as (-infinity, infinity). This is because the graph approaches negative infinity as x approaches -4 and goes up towards positive infinity as x increases.

Summary & Key Takeaways

• The video discusses finding the domain of the function f(x) = Ln(x) + 4 by setting the inside of the logarithm greater than 0.

• It explains how to graph the function by selecting various x-values and solving for corresponding y-values.

• The video also covers the range and asymptotes of the function, as well as finding the inverse by switching x and y and isolating y.