Exponential Functions Problem, find k and a in f(x)=ka^(−x)

In summary, the conversation involves finding the values of k and a in the given function, solving for x1, and showing the relationship between an increase in x and the corresponding decrease in the value of the function. The solution for k is 8 and a is -√2. To find x1, the equation 8a^-x1 = 4 is solved, resulting in x1 = 2. To show the relationship in (3), the function with x+2 substituted for x is used to calculate the ratio of the values of f, resulting in a reduction of 50%.
  • #1
Yankel
395
0
Hello all,

I am trying to solve the following problem:

In the given graph, we see the function:

\[f(x)=ka^{-x} , x\geq 0\]

View attachment 8053

1) Find k and a

2) Find x1

3) Show that an increase of 2 units in x brings a 50% reduction in the value of the function f.

I have tried solving it, but taking two known points from the graph and putting them in the function. I got immediately that:

\[k=8\]

and than less clear, that:

\[a=-\sqrt{2}\]

In the matter of fact, I got two solutions, a solution without the minus sign was there too, but it didn't match the given points...so in order to solve (2), I just guessed and checked my guess.

I am not sure how to show number (3).

Can you kindly verify that my solution is correct and assist with section 3 ? What am I missing with the multiple solutions for a ?

Thank you in advance !
 

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  • #2
Hi Yankel.

1) $k=8$ is correct. To find $a$, solve $8a^{-4}=2$ for $a$. Can you continue? (Hint: Choose the positive root.)

2) Solve $8a^{-x_1}=4$ for $x_1$. Okay?

3) We'll complete this after the two exercises above. :)
 
  • #3
Hi,

If I am not mistaken a is equal to the square root of 2.

Then x1 is 2. Correct ?
 
  • #4
That's correct. :)
 
  • #5
So...how do we continue with number 3 then ? :-)

I took the function we get, put x+2 instead of x and divided the value of f with x by the value of f with x+2. Got 2. Is if sufficient ?
 
  • #6
Certainly is. Good work! :D
 

FAQ: Exponential Functions Problem, find k and a in f(x)=ka^(−x)

1. What is an exponential function?

An exponential function is a mathematical function in which the independent variable appears as an exponent. It is commonly written in the form f(x) = ka^x, where k is a constant and a is the base.

2. How do you solve for k and a in an exponential function?

To solve for k and a in an exponential function, you need to have two sets of coordinates (x, y) from the function. Then, you can use the formula y = ka^x to create a system of equations and solve for k and a. Alternatively, you can use logarithms to solve for k and a.

3. What is the significance of k and a in an exponential function?

In an exponential function, k is the initial value or starting point, and a is the rate of change or growth factor. K determines where the graph of the function intersects with the y-axis, and a determines the steepness of the curve.

4. Can an exponential function have a negative base?

No, an exponential function cannot have a negative base. The base must be a positive number, and the exponent can be any real number. However, if the exponent is a negative number, the function will have an inverse relationship and will approach the x-axis as x increases.

5. How are exponential functions used in real life?

Exponential functions are used to model many real-life situations, such as population growth, compound interest, and radioactive decay. They are also commonly used in fields such as economics, biology, and physics to study growth and decay processes.

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