High School

If [tex]f(x)[/tex] is an exponential function where [tex]f(1) = 8[/tex] and [tex]f(5.5) = 69[/tex], then find the value of [tex]f(8.5)[/tex] to the nearest hundredth.

Answer :

The value of f(8.5) is 172.32.

Finding f(8.5) for the given exponential function:

We know that f(x) is an exponential function, but its exact form is unknown. However, we have two data points to work with:

f(1) = 8

f(5.5) = 69

The general form of an exponential function is:

f(x) = a * b^x

where:

a is the initial value (y-intercept)

b is the growth factor (base raised to the power of x)

1. Find the initial value (a)

Using the point f(1) = 8, we can plug in x = 1 and y = 8 into the general equation:

8 = a * b^1

8 = a * b

Since b is unknown, we cannot directly solve for a. However, we can keep this equation for later.

2. Find the growth factor (b)

Using the point f(5.5) = 69, we can plug in x = 5.5 and y = 69 into the general equation:

69 = a * b^5.5

Divide both sides by a (which we don't know yet, but it's not zero since f(1) is not zero):

69/a = b^5.5

Now, we can use the equation we obtained in step 1:

8 = a * b

Substitute b from this equation into the equation above:

69/(8) = (a * b)^5.5

Simplify:

69/8 = (8 * b)^5.5

8.625 = 32 * b^5.5

Take the fifth root of both sides:

(8.625)^(1/5.5) = (32 * b^5.5)^(1/5.5)

b ≈ 1.2247

3. Find f(8.5)

Now that we know b ≈ 1.2247, we can plug it back into the general equation along with the initial value from step 1 (a = 8):

f(8.5) = 8 * (1.2247)^(8.5)

≈ 172.32 (rounded to two decimal places)

Therefore, f(8.5) ≈ 172.32.

The answer is f(8.5) ≈ 172.32.

Finding f(8.5) for the given exponential function:

We know that f(x) is an exponential function, but its exact form is unknown. However, we have two data points to work with:

f(1) = 8

f(5.5) = 69

The general form of an exponential function is:

f(x) = a * b^x

where:

a is the initial value (y-intercept)

b is the growth factor (base raised to the power of x)

1. Find the initial value (a)

Using the point f(1) = 8, we can plug in x = 1 and y = 8 into the general equation:

8 = a * b^1

8 = a * b

Since b is unknown, we cannot directly solve for a. However, we can keep this equation for later.

2. Find the growth factor (b)

Using the point f(5.5) = 69, we can plug in x = 5.5 and y = 69 into the general equation:

69 = a * b^5.5

Divide both sides by a (which we don't know yet, but it's not zero since f(1) is not zero):

69/a = b^5.5

Now, we can use the equation we obtained in step 1:

8 = a * b

Substitute b from this equation into the equation above:

69/(8) = (a * b)^5.5

Simplify:

69/8 = (8 * b)^5.5

8.625 = 32 * b^5.5

Take the fifth root of both sides:

(8.625)^(1/5.5) = (32 * b^5.5)^(1/5.5)

b ≈ 1.2247

3. Find f(8.5)

Now that we know b ≈ 1.2247, we can plug it back into the general equation along with the initial value from step 1 (a = 8):

f(8.5) = 8 * (1.2247)^(8.5)

≈ 172.32 (rounded to two decimal places)

Therefore, f(8.5) ≈ 172.32.