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Math  /  Algebra

QuestionFind a simplified formula for the power function whose values are represented in the table below. \begin{tabular}{|c|c|c|c|c|} \hlinex=x= & 2 & 3 & 4 & 5 \\ \hlinef(x)=f(x)= & 32 & 108 & 256 & 500 \\ \hline \end{tabular} f(x)=f(x)= □\square help (formulas)

Studdy Solution

STEP 1

1. The function f(x) f(x) is a power function, meaning it can be expressed in the form f(x)=axb f(x) = ax^b .
2. We are given specific values of f(x) f(x) for certain values of x x .

STEP 2

1. Identify the form of the power function.
2. Use given data points to create equations.
3. Solve the equations to find the constants a a and b b .
4. Write the simplified formula for f(x) f(x) .

STEP 3

Assume the power function is of the form f(x)=axb f(x) = ax^b .

STEP 4

Use the data point (2,32) (2, 32) to form the equation: 32=a⋅2b 32 = a \cdot 2^b
Use the data point (3,108) (3, 108) to form the equation: 108=a⋅3b 108 = a \cdot 3^b

STEP 5

Divide the second equation by the first to eliminate a a : 10832=a⋅3ba⋅2b \frac{108}{32} = \frac{a \cdot 3^b}{a \cdot 2^b}
Simplify to find b b : 10832=(32)b \frac{108}{32} = \left(\frac{3}{2}\right)^b
Calculate: 10832=278=(32)b \frac{108}{32} = \frac{27}{8} = \left(\frac{3}{2}\right)^b
Take the logarithm of both sides: log⁡(278)=b⋅log⁡(32) \log\left(\frac{27}{8}\right) = b \cdot \log\left(\frac{3}{2}\right)
Solve for b b : b=log⁡(278)log⁡(32)=3 b = \frac{\log\left(\frac{27}{8}\right)}{\log\left(\frac{3}{2}\right)} = 3

STEP 6

Substitute b=3 b = 3 back into the equation 32=a⋅2b 32 = a \cdot 2^b : 32=a⋅23 32 = a \cdot 2^3
Solve for a a : 32=a⋅8 32 = a \cdot 8 a=4 a = 4

STEP 7

The simplified formula for the power function is: f(x)=4x3 f(x) = 4x^3
The simplified formula for f(x) f(x) is:
4x3 \boxed{4x^3}

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