Problem 76 $f(x)=\ln (x+2)$... [FREE SOLUTION]

91影视

Trigonometry

Ron Larson, Robert Hostetler

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 5: Problem 76

$f(x)=\ln (x+2)$

Short Answer

Expert verified

The function $f(x)=\ln (x+2)$ is a shift of the natural logarithm function 2 units to the left. The domain is $x>-2$ and the range is all real numbers. As $x$ approaches -2, $f(x)$ decreases without bound; as $x\rightarrow \infty$, $f(x)$ increases and goes to $\infty$.

Step by step solution

Understand the function

The first step is to understand the function given, which is $f(x)=\ln (x+2)$. In this function, $x+2$ is the argument of the natural logarithm function. The graph of this function would be identical to the graph of $\ln (x)$ but shifted 2 units to the left.

Identify the domain and range

Next, find the domain and range of the function. Recall that for a natural logarithmic function $y=\ln(x)$, the domain is $x>0$ and the range is all real numbers. So, for our function, $f(x)=\ln (x+2)$, the domain will be $x>-2$ and the range is again all real numbers.

Describe the behavior of the function

A logarithmic function of the form $y = ln(x)$, where $x + c$ (in this case, it is $x + 2$), will shift to left by c units. Hence, in this case it shifts 2 units to the left. As $x$ approaches -2, $f(x)$ will decrease without bound which means it goes to negative infinity. As $x$ goes to positive infinity, $f(x)$ will increase and goes to positive infinity. This simply means that $f(x)$ will rise on the right and fall on the left.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain and Range

The logarithmic function presented here is $f(x) = \ln(x+2)$. One of the key aspects of logarithmic functions is understanding their domain and range. In simpler terms, the domain comprises the set of input values for which the function is defined, whereas the range refers to all possible output values.

For the natural logarithmic function $\ln(x)$, the domain is $x > 0$, which means only positive values for $x$ are allowed.
When we consider $\ln(x+2)$, the domain shifts to $x > -2$, because whatever the argument of the logarithm function (in this case $x+2$) must be greater than zero.
Hence, the function is only valid for values of $x$ greater than $-2$.

As for the range of $f(x) = \ln(x+2)$, like all logarithmic functions, it adopts any real number from negative to positive infinity. This implies that, irrespective of how large or small the result from the logarithm operation is, the output ($y$) can be any real number.

Function Transformation

In function transformation, it's pivotal to recognize how changes affect a function's graph. The primary function we analyze is by starting with $y = \ln(x)$. When we express the function as $f(x) = \ln(x+2)$, our graph undergoes a transformation.

Adding 2 to $x$ shifts the graph horizontally. Specifically, $x+2$ results in a 2-unit shift to the left.
This occurs because each input to $x$ in $\ln(x+2)$ raises the argument of the logarithm to positive numbers earlier than in $\ln(x)$.

Thus, understanding transformations such as horizontal shifts help predict how the original graph rearranges. The graph remains unchanged in shape and scale; it merely slides along the x-axis.

Graph Behavior

The graph of a logarithmic function like $f(x)=\ln(x+2)$ exhibits distinct behavior patterns. Recognizing these patterns is crucial when analyzing its graph.

As $x$ nears $-2$, $f(x)$ experiences a steep decline towards negative infinity. This is because the logarithm becomes infinitely negative just before the value making the function undefined.
Conversely, as $x$ shoots off towards positive infinity, $f(x)$ gradually climbs upwards towards infinity. There's no upper bound as $x$ grows larger.
This behavior creates a graph that rises on the right and falls sharply on the left, with a vertical asymptote at $x = -2$, delineating where $f(x)$ is not defined.

Logarithmic functions tend to rise slowly as $x$ increases because each additional unit increment to the input translates into smaller logarithmic output deltas.

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91影视

\(f(x)=\ln (x+2)\)

Short Answer

Step by step solution

Understand the function

Identify the domain and range

Describe the behavior of the function

Key Concepts

Domain and Range

Function Transformation

Graph Behavior

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