Problem 46 Use transformations to graph eac... [FREE SOLUTION]

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Chapter 6: Problem 46

Use transformations to graph each function. Determine the domain, range, horizontal asymptote, and y-intercept of each function. $$ f(x)=3^{x}-2 $$

Short Answer

Expert verified

Domain: (-鈭�, 鈭�), Range: (-2, 鈭�), Horizontal Asymptote: y = -2, Y-intercept: (0, -1)

Step by step solution

- Identify the Base Function

Identify the base exponential function. The base function for the given equation is 3^x.

- Vertical Shift

Recognize and apply the vertical shift. The function f(x) has a vertical shift of -2. This means the graph of 3^x is shifted downward by 2 units.

- Horizontal Asymptote

Determine the horizontal asymptote. For the function f(x) = 3^x - 2, the horizontal asymptote is y = -2.

- Y-intercept

Find the y-intercept by substituting x = 0 into the function. So, f(0) = 3^0 - 2, which simplifies to 1 - 2 = -1. Therefore, the y-intercept is (0, -1).

- Domain

The domain of an exponential function is all real numbers. Therefore, the domain of f(x) = 3^x - 2 is (-鈭�, 鈭�).

- Range

Determine the range of the function. Since the graph of 3^x is shifted downward by 2 units, the range is all real numbers greater than -2. Therefore, the range is (-2, 鈭�).

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Key Concepts

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

exponential function

An exponential function is a type of function where the variable is an exponent. These functions have the form $ f(x) = a \times b^x \, \text{or} \, f(x) = b^x $, where $a$ and $b$ are constants. In the context of the given function $ f(x) = 3^x - 2 $, the base function is $3^x$. This means our exponential function will grow as $x$ increases, rapidly rising towards positive infinity.

Key points to remember about exponential functions:

They are not linear; they grow or decay at an increasing rate.
When the base $b > 1$, the function represents growth.
For $b < 1$ (but $b > 0$), the function represents decay.
The graph always passes through the point (0,1) if there is no vertical shift.

vertical shift

A vertical shift occurs when the entire graph of the function is moved up or down along the y-axis. In our function $f(x) = 3^x - 2$, there is a vertical shift of -2. This means the graph of the base function $3^x$ is shifted downward by 2 units.

Here's what you need to know about vertical shifts:

If the transformation is $ f(x) = g(x) + k $, where $k > 0$, the graph shifts upwards.
For $ f(x) = g(x) - k $, where $k > 0$, the graph shifts downwards.
This vertical shift impacts the y-intercept and the horizontal asymptote.

Understanding and identifying this shift helps in accurately plotting the graph and determining its key characteristics, such as domain and range.

horizontal asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never actually touches as $ x $ goes to positive or negative infinity. For the function $f(x) = 3^x - 2$, the horizontal asymptote is $ y = -2 $.

Important points about horizontal asymptotes:

They indicate the behavior of the function at extreme values of $ x $.
For $f(x) = 3^x - 2$, the base function $3^x$ normally approaches $ y = 0 $. The subtraction of 2 shifts this asymptote down to $ y = -2 $.
This line helps in understanding the limiting value of the function as $ x $ grows very large or very small.

domain and range

The domain of a function describes all the possible input values ($x$) that the function can accept, while the range describes all possible output values ($y$).

Let's explore the domain and range for $f(x) = 3^x - 2$:

**Domain:** Exponential functions typically accept all real numbers for $x$. So, the domain of $f(x)$ is $( -鈭�, 鈭� )$.
**Range:** To determine the range, consider the vertical shift. The base function $3^x$ has a range of $( 0, 鈭� )$, but shifting it down by 2 units changes the range. Thus, the range of $f(x) = 3^x - 2$ is $( -2, 鈭� )$.

Understanding the domain and range is crucial for fully grasping how the function behaves across different values of $x$ and predicting its output values for any input within its domain.

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91影视

Use transformations to graph each function. Determine the domain, range, horizontal asymptote, and y-intercept of each function. $$ f(x)=3^{x}-2 $$

Short Answer

Step by step solution

- Identify the Base Function

- Vertical Shift

- Horizontal Asymptote

- Y-intercept

- Domain

- Range

Key Concepts

exponential function

vertical shift

horizontal asymptote

domain and range

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