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, 0, point, 5, equals, 16f(0.5)=16 and f, of, 7, equals, 38f(7)=38, then find the value of f, of, 1f(1), to the nearest hundredth.

We are given that \( f(x) \) is an exponential function of the form:

\[
f(x) = ab^x
\]

where \( f(0.5) = 16 \) and \( f(7) = 38 \). We need to find \( f(1) \).

### Step 1: Express the function using the given form
The general form of the exponential function is:

\[
f(x) = ab^x
\]

We can use the two known points \( f(0.5) = 16 \) and \( f(7) = 38 \) to form two equations.

1. At \( x = 0.5 \), we have:
\[
f(0.5) = ab^{0.5} = 16
\]
2. At \( x = 7 \), we have:
\[
f(7) = ab^7 = 38
\]

### Step 2: Solve for \( a \) and \( b \)
We now have two equations:
1. \( ab^{0.5} = 16 \)
2. \( ab^7 = 38 \)

To eliminate \( a \), divide the second equation by the first equation:

\[
\frac{ab^7}{ab^{0.5}} = \frac{38}{16}
\]

Simplifying:

\[
b^{7 - 0.5} = \frac{38}{16} = 2.375
\]

This gives:

\[
b^{6.5} = 2.375
\]

To solve for \( b \), take the 6.5th root of both sides:

\[
b = \left( 2.375 \right)^{\frac{1}{6.5}} \approx 1.138
\]

### Step 3: Solve for \( a \)
Now that we have \( b \approx 1.138 \), substitute it back into the first equation:

\[
ab^{0.5} = 16
\]

\[
a \times 1.138^{0.5} = 16
\]

Simplifying \( 1.138^{0.5} \approx 1.067 \), we get:

\[
a \times 1.067 = 16
\]

Solving for \( a \):

\[
a \approx \frac{16}{1.067} \approx 15.0
\]

### Step 4: Find \( f(1) \)
Now that we have \( a \approx 15.0 \) and \( b \approx 1.138 \), substitute into the function \( f(x) = ab^x \) to find \( f(1) \):

\[
f(1) = 15.0 \times 1.138^1 = 15.0 \times 1.138 \approx 17.07
\]

### Final Answer:
The value of \( f(1) \) to the nearest hundredth is:

\[
f(1) \approx 17.07
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What would happen if \( f(x) \) were a linear function instead of exponential?
2. How can the rate of growth of an exponential function be interpreted from \( b \)?
3. What if we knew \( f(x) \) at three points? How would that affect the process?
4. Can you find the derivative of \( f(x) = ab^x \) with respect to \( x \)?
5. How does the base \( b \) relate to exponential growth or decay?

**Tip**: When solving for unknown parameters in exponential functions, using logarithms can help simplify the calculations involving powers.

If f(x) is an exponential function of the form y = ab^x where f(0.5) = 16 and f(7) = 38, then find the value of f(1) to the nearest hundredth.

Find the value of f(1) for an exponential function of the form f(x) = ab^x, where f(0.5) = 16 and f(7) = 38. This problem covers solving exponential equations by determining the constants a and b, then calculating f(1). Suitable for students in Grades 9-12.

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