Find a formula for the exponential function which satisfies the given conditions:

f(3)=1/4 and f(−1)=4.

Exponential function:

f(x) =a bˣ


f(3) = a b³ = 1 / 4

f(-1) = a b⁻¹ = 4

a b³ = 1 / 4
÷
a b⁻¹ = 4
__________

b⁴ = 1 / 16

b = ∜1/16

b = 1 / 2

Substitute that into:

a b³ = 1 / 4

a ( 1 / 2 )³ = 1 / 4

a ∙ ( 1 / 8 ) = 1 / 4

Divide both sides by 1 / 8

a = ( 1 / 4 ) / ( 1 / 8 )

a = 2

Your exponential function:

f(x) = a bˣ

f(x) = 2 ∙ ( 1 / 2 )ˣ

Proof:

x = 3

f(3) = 2 ∙ ( 1 / 2 )³ = 2 ∙ ( 1 / 8 ) = 1 / 4

x = - 1

f(-1) = 2 ∙ ( 1 / 2 ) ⁻¹ = 2 / ( 1 / 2 ) = 4

Why did the exponential function break up with its old girlfriend? Because it wanted to explore new exponential growth!

So, let's find the formula for this exponential function, shall we?

An exponential function can be represented as f(x) = a * b^x, where a is a constant and b is the base of the exponential function.

Given the conditions f(3) = 1/4 and f(-1) = 4, we can set up two equations:

1/4 = a * b^3 (Equation 1)
4 = a * b^(-1) (Equation 2)

To solve for a and b, let's manipulate the second equation first to get rid of the negative exponent:

4 = a * (1/b)

Now, let's substitute this into Equation 1:

1/4 = (4 * b) * b^3

Alright, now we can solve for b:

1/4 = 4b^4

b^4 = 1/16

Taking the fourth root on both sides, we get:

b = 1/2

Now, we can substitute the value of b into Equation 2 to solve for a:

4 = a * (1/2)^(-1)

4 = a * 2

a = 2

Therefore, the formula for the exponential function that satisfies the given conditions is:

f(x) = 2 * (1/2)^x

Hope that brings a smile to your face, just like the exponential function finding new growth!

To find the formula for the exponential function that satisfies the given conditions, we can use the general form of an exponential function:

f(x) = a * b^x

where "a" and "b" are constants.

We are given two points: (3, 1/4) and (-1, 4).

Using the point (3, 1/4), we can substitute the x and y values into the formula:

1/4 = a * b^3

Using the point (-1, 4), we can substitute the x and y values into the formula:

4 = a * b^(-1)

We now have a system of equations:

1/4 = a * b^3
4 = a * b^(-1)

To solve for "a" and "b," we can take the ratio of the two equations:

(1/4) / 4 = (a * b^3) / (a * b^(-1))

1/16 = b^4 / b^(-1)

1/16 = b^(4 + 1)

1/16 = b^5

To solve for "b," we can take the fifth root of both sides:

b = (1/16)^(1/5)

b = 1/2

Now that we have the value of "b," we can substitute it back into one of the original equations to solve for "a." Let's use the equation 4 = a * b^(-1):

4 = a * (1/2)^(-1)
4 = a * 2

Solving for "a," we get:

a = 4/2
a = 2

Thus, the formula for the exponential function that satisfies the given conditions is:

f(x) = 2 * (1/2)^x

To find a formula for the exponential function that satisfies the given conditions, we need to use the general form of an exponential function, which is: f(x) = a * b^x, where "a" and "b" are constants.

Given that f(3) = 1/4, we can substitute these values into the equation:

1/4 = a * b^3.

Similarly, for f(-1) = 4:

4 = a * b^(-1).

We now have a system of two equations with two variables (a and b). To solve for a and b, we can set up a ratio by dividing the first equation by the second equation:

(1/4) / 4 = (a * b^3) / (a * b^(-1)).

This simplifies to:

1/16 = b^4.

To solve for b, take the fourth root of both sides:

b = ±√(1/16) = ±(1/2).

Now that we have the possible values for b, we can substitute one of them back into one of the initial equations to solve for a. Let's use b = 1/2:

4 = a * (1/2)^(-1).

This simplifies to:

4 = a * 2.

Dividing both sides by 2 gives us:

2 = a.

Therefore, one possible formula for the exponential function that satisfies the given conditions is:

f(x) = 2 * (1/2)^x.

Similarly, using b = -1/2 would give us another possible formula:

f(x) = 2 * (-1/2)^x.