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For the following function, briefly describe how the graph can be obtained from the graph of a basic logarithmic function. Then, graph the function and state the domain and the vertical asymptote. f(x) = 7 - In x Describe how the graph of f(x) can be obtained from the graph of a basic logarithmic function. The graph of f(x) = 7 - In x is a transformation of the graph of f(x) = In x by a reflection across the and then a translation units. Use the graphing tool to graph the equation.

User Jason Lin
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Answer

1) Graph is shown below in the 'Explanation'.

2) Domain: x > 0

In interval notation,

Domain: (0, ∞)

3) Vertical asymptote: x = 0

Horizontal asymptote: y = 7

4) The transformations required to turn f(x) = In x into f(x) = 7 - In x include

A reflection of f(x) = In x about the x-axis.

Then, this reflected image is then translated 7 units upwards.

Step-by-step explanation

The graph of function is attached below

For the domain and asymptote,

Domain

The domain of a function refers to the values of the independent variable (x), where the dependent variable [y or f(x)] or the function has a corresponding real value. The domain is simply the values of x for which the output also exists. It is the region around the x-axis that the graph of the function spans.

We know that the logarithm of a number only exists if the number is positive.

So,

Domain: x > 0

In interval notation,

Domain: (0, ∞)

Asymptote

Asymptotes are the points on either the x-axis or the y-axis where the graph of the function doesn't touch.

They are usually denoted by broken lines.

For this question, we know that the value of f(x) cannot go beyond f(x) = 7 and x = 0

Vertical asymptote: x = 0

Horizontal asymptote: y = 7

For the transformation

When a function f(x) is translated horizontally along the x-axis by a units, the new function is represented as

f(x + a) when the translation is by a units to the left.

f(x - a) when the translation is by a units to the right.

When a function f(x) is translated vertically along the y-axis by b units, the new function is represented as

f(x) + b when the translation is by b units upwards.

f(x) - b when the translation is by b units downwards.

So, if the original function is

f(x) = In x

f(x) = -In x

This reflects the original function about the x-axis.

Then,

f(x) = 7 - In x

This translates the reflected function by 7 units upwards.

For the following function, briefly describe how the graph can be obtained from the-example-1
User Rmw
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