If f, of, xf(x) is an exponential function of the form of y, equals, a, b, to the power xy=ab 
x
  where f, of, 3, point, 5, equals, 16f(3.5)=16 and f, of, 7, equals, 62f(7)=62, then find the value of f, of, 10, point, 5f(10.5), to the nearest hundredth.

We are given that the function \( f(x) \) is an exponential function of the form \( f(x) = ab^x \), and we know two values of the function:

\[
f(3.5) = 16 \quad \text{and} \quad f(7) = 62.
\]

We are asked to find \( f(10.5) \) to the nearest hundredth.

### Step 1: Write the general form
The exponential function is given by \( f(x) = ab^x \), where \( a \) and \( b \) are constants to be determined.

### Step 2: Set up equations using known values
We know:

1. \( f(3.5) = ab^{3.5} = 16 \)
2. \( f(7) = ab^7 = 62 \)

These give us the system of equations:

\[
ab^{3.5} = 16 \quad \text{(Equation 1)}
\]
\[
ab^7 = 62 \quad \text{(Equation 2)}
\]

### Step 3: Solve for \( b \)
We divide Equation 2 by Equation 1 to eliminate \( a \):

\[
\frac{ab^7}{ab^{3.5}} = \frac{62}{16}
\]

Simplifying:

\[
b^{7 - 3.5} = \frac{62}{16}
\]

\[
b^{3.5} = \frac{62}{16} = 3.875
\]

Now, solve for \( b \) by taking the cube root:

\[
b = (3.875)^{\frac{1}{3.5}}
\]

### Step 4: Solve for \( a \)
Once we have \( b \), we can substitute it back into one of the original equations (say, Equation 1) to find \( a \).

### Step 5: Find \( f(10.5) \)
Finally, using the values of \( a \) and \( b \), we can find \( f(10.5) \) by substituting \( x = 10.5 \) into the equation \( f(x) = ab^x \).

I'll calculate this step by step.The value of \( f(10.5) \) is approximately 240.25, to the nearest hundredth.

Would you like further details or have any questions about the steps? Here are some related questions to explore:

1. What happens to an exponential function as \( x \) increases indefinitely?
2. How does the base \( b \) of an exponential function affect the function's growth rate?
3. Can exponential functions ever have negative values for \( f(x) \)?
4. How would you find the inverse of an exponential function?
5. What are some real-life applications of exponential growth or decay?

**Tip**: For exponential functions, the base \( b \) determines how quickly the function grows (if \( b > 1 \)) or decays (if \( 0 < b < 1 \)).

If f(x) is an exponential function of the form y = ab^x where f(3.5) = 16 and f(7) = 62, then find the value of f(10.5), to the nearest hundredth.

Learn how to solve for f(10.5) in an exponential function of the form f(x) = ab^x, given f(3.5) = 16 and f(7) = 62. This problem involves exponential growth, logarithmic calculations, and algebraic manipulation. Suitable for high school students in Grades 9-12.

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