Questions: Graph. f(x) = 3/2 x

Transcript text: Graph. \[ f(x)=\frac{3}{2} x \]

Solution

Solution Steps

Step 1: Identify the Function Type

The given function is \( f(x) = \left(\frac{3}{2}\right)^x \). This is an exponential function of the form \( f(x) = a^x \), where \( a = \frac{3}{2} \).

Step 2: Determine Key Points

To graph the function, determine key points by substituting different values of \( x \):

For \( x = -2 \): \( f(-2) = \left(\frac{3}{2}\right)^{-2} = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \approx 0.44 \)
For \( x = -1 \): \( f(-1) = \left(\frac{3}{2}\right)^{-1} = \frac{2}{3} \approx 0.67 \)
For \( x = 0 \): \( f(0) = \left(\frac{3}{2}\right)^0 = 1 \)
For \( x = 1 \): \( f(1) = \left(\frac{3}{2}\right)^1 = \frac{3}{2} = 1.5 \)
For \( x = 2 \): \( f(2) = \left(\frac{3}{2}\right)^2 = \frac{9}{4} = 2.25 \)

Step 3: Plot the Points

Plot the points on the graph:

\((-2, 0.44)\)
\((-1, 0.67)\)
\((0, 1)\)
\((1, 1.5)\)
\((2, 2.25)\)

Step 4: Draw the Exponential Curve

Connect the plotted points with a smooth curve to represent the exponential growth of the function \( f(x) = \left(\frac{3}{2}\right)^x \).

Final Answer

The graph of the function \( f(x) = \left(\frac{3}{2}\right)^x \) is an exponential curve that passes through the points \((-2, 0.44)\), \((-1, 0.67)\), \((0, 1)\), \((1, 1.5)\), and \((2, 2.25)\). The curve increases rapidly as \( x \) increases.

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