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Use transformations to graph each function. Determine the domain, range, horizontal asymptote, and y-intercept

1. f(x) = 3³ - 2
2. f(x) = 3-1
solve each equation.
3. 3 = 81
4. 4*.2*² = 16²
5. (e¹)¹. e²² = e¹2​

User Gasolin
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1 Answer

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Let's address each part of the question:

1. \( f(x) = 3^x - 2 \)

This function is a transformation of the exponential function \( f(x) = 3^x \). It is shifted down by 2 units due to the "- 2" outside the exponent.

- Domain: The domain of exponential functions is always all real numbers.

- Range: Since the base is \( 3^x \), the range will be all positive real numbers shifted down by 2 units.

- Horizontal Asymptote: There's no horizontal asymptote in exponential functions.

- Y-intercept: To find the y-intercept, set \( x = 0 \) and solve for \( f(0) \).

\( f(0) = 3^0 - 2 = 1 - 2 = -1 \)

2. \( f(x) = 3^{-1} \)

This function is a constant function where \( f(x) \) is always equal to \( \frac{1}{3} \).

- Domain: All real numbers.

- Range: The range is just the constant value \( \frac{1}{3} \).

- Horizontal Asymptote: There's no horizontal asymptote in constant functions.

- Y-intercept: The y-intercept is where \( x = 0 \), so \( f(0) = \frac{1}{3} \).

Now, let's solve the equations:

3. \( 3 = 81 \)

This equation is not true. There is no solution since \( 3 \) does not equal \( 81 \).

4. \( 4 \cdot 0.2 \cdot 2^2 = 16^2 \)

Let's solve this:

\( 4 \cdot 0.2 \cdot 2^2 = 16^2 \)

\( 4 \cdot 0.2 \cdot 4 = 256 \)

\( 3.2 \cdot 4 = 256 \)

\( 12.8 = 256 \)

This equation is not true. There is no solution since \( 12.8 \) does not equal \( 256 \).

5. \( (e^1)^1 \cdot e^{22} = e^{12} \)

Let's solve this:

\( (e^1)^1 \cdot e^{22} = e^{12} \)

\( e \cdot e^{22} = e^{12} \)

\( e^{23} = e^{12} \)

Now, since the bases are the same, we equate the exponents:

\( 23 = 12 \)

This equation is not true. There is no solution since \( 23 \) does not equal \( 12 \).

User Ggarber
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