To find the equation of the exponential function in the form \( f(x) = a(b)^x \), we need to determine the values of \( a \) and \( b \) using the given points \((0, -2)\) and \((1, -3)\).
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Using the point \((0, -2)\): \[ f(0) = a(b)^0 = a \cdot 1 = a \] Therefore, we have: \[ a = -2 \]
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Using the point \((1, -3)\): \[ f(1) = a(b)^1 = a \cdot b \] Plugging in the values, we get: \[ -3 = -2b \] Solving for \( b \): \[ -3 = -2b \implies b = \frac{-3}{-2} = \frac{3}{2} \]
Now we can write the equation of the exponential function: \[ f(x) = -2\left(\frac{3}{2}\right)^x \]
Thus, the equation is: \[ f(x) = -2\left(\frac{3}{2}\right)^x \]
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