let f(x)=mx^2 + 2x + 5 and g(x)= 2x^2 - nx - 2. the functions are combined to form the new function h(x)= f(x)*g(x). points (1,-40) and (-1,24) satisfy the new function. determine f(x) and g(x)
Well, to solve this problem, let's start by substituting the given points into the new function h(x).
When we substitute (1, -40) into h(x), we get:
h(1) = f(1) * g(1)
And since this point satisfies the new function, we know that h(1) = -40. So now our equation becomes:
-40 = f(1) * g(1)
Similarly, when we substitute (-1, 24) into h(x), we get:
h(-1) = f(-1) * g(-1)
And since this point satisfies the new function, we know that h(-1) = 24. So now our equation becomes:
24 = f(-1) * g(-1)
Now, let's solve these two equations simultaneously to find f(x) and g(x).
Since f(x) and g(x) are quadratic functions, and h(x) is formed by their product, we can write them in the general quadratic form:
f(x) = a1x^2 + b1x + c1
g(x) = a2x^2 + b2x + c2
Now let's substitute x = 1 into these equations:
-40 = a1 + b1 + c1 ...(1)
Next, substitute x = -1:
24 = a1 - b1 + c1 ...(2)
By solving equations (1) and (2), we can find the values of a1, b1, and c1 which represent f(x).
By the same method, we can find a2, b2, and c2 which represent g(x).
I hope this clears things up for you!
To determine the functions f(x) and g(x), we will use the points (1, -40) and (-1, 24) that satisfy the new function h(x) = f(x) * g(x).
Step 1: Let's start by finding f(x):
- We'll plug the point (1, -40) into the equation h(x) = f(x) * g(x), where x = 1.
- h(1) = f(1) * g(1), and h(1) = -40.
- Substitute h(1) = -40 into the equation: -40 = f(1) * g(1).
- This gives us the first equation: f(1) * g(1) = -40.
Step 2: Now let's find g(x):
- Similarly, we'll plug the point (-1, 24) into the equation h(x) = f(x) * g(x), where x = -1.
- h(-1) = f(-1) * g(-1), and h(-1) = 24.
- Substitute h(-1) = 24 into the equation: 24 = f(-1) * g(-1).
- This gives us the second equation: f(-1) * g(-1) = 24.
Step 3: Solve the system of equations:
- We can use the method of substitution to solve this system of equations.
- Divide the first equation (f(1) * g(1) = -40) by the second equation (f(-1) * g(-1) = 24):
(f(1) * g(1))/(f(-1) * g(-1)) = (-40)/(24).
- Simplifying this gives: (f(1)/f(-1)) * (g(1)/g(-1)) = -5/3.
Step 4: Determine f(x) and g(x):
- Notice that the ratio of f(1)/f(-1) and g(1)/g(-1) must be equal to -5/3 for any combination of f(x) and g(x) that satisfy the given conditions.
- We need to find values of f(1)/f(-1) and g(1)/g(-1) such that their ratio equals -5/3.
There are different possibilities for f(x) and g(x) that can satisfy the given conditions. Here are two possible solutions:
Solution 1:
- Let f(1)/f(-1) = -5 and g(1)/g(-1) = 3.
- Multiply both equations by the denominators to find the values of f(1) and g(1):
-5f(-1) = f(1) and 3g(-1) = g(1).
- Therefore, f(x) = -5x and g(x) = 3x.
Solution 2:
- Let f(1)/f(-1) = 5 and g(1)/g(-1) = -3.
- Multiply both equations by the denominators to find the values of f(1) and g(1):
5f(-1) = f(1) and -3g(-1) = g(1).
- Therefore, f(x) = 5x and g(x) = -3x.
Please note that there may be other combinations that satisfy the given conditions, but these are two possible solutions.
To determine f(x) and g(x), we need to use the given information about the points that satisfy the new function h(x) = f(x) * g(x).
Let's start by substituting the point (1, -40) into the equation h(x) = f(x) * g(x):
h(1) = f(1) * g(1)
-40 = f(1) * g(1)
Similarly, substituting the point (-1, 24) into the equation:
h(-1) = f(-1) * g(-1)
24 = f(-1) * g(-1)
We have two equations now:
-40 = f(1) * g(1)
24 = f(-1) * g(-1)
We can rewrite f(x) and g(x) in terms of their coefficients:
f(x) = mx^2 + 2x + 5
g(x) = 2x^2 - nx - 2
Substituting x = 1 into the equations:
f(1) = m(1)^2 + 2(1) + 5
f(1) = m + 2 + 5
f(1) = m + 7
g(1) = 2(1)^2 - n(1) - 2
g(1) = 2 - n - 2
g(1) = -n
Substituting x = -1 into the equations:
f(-1) = m(-1)^2 + 2(-1) + 5
f(-1) = m + (-2) + 5
f(-1) = m + 3
g(-1) = 2(-1)^2 - n(-1) - 2
g(-1) = 2 + n - 2
g(-1) = n
Now we can substitute these results back into our original equations:
-40 = (m + 7) * (-n)
24 = (m + 3) * n
Rearranging the equations:
-40 = -mn - 7n
24 = mn + 3n
Dividing the first equation by -n, we get:
40 = mn + 7
Rearranging the second equation:
24 - 3n = mn
From equation 1, we have:
40 = mn + 7
mn = 40 - 7
mn = 33
Substituting in equation 2:
24 - 3n = 33
-3n = 33 - 24
-3n = 9
n = -3
Now, substitute the value of n = -3 back into equation 1:
mn = 33
m(-3) = 33
-3m = 33
m = -11
Hence, f(x) = -11x^2 + 2x + 5 and g(x) = 2x^2 + 3x - 2.
use the two points they gave you.
f(1) = m+2+5 = m+7
g(1) = 2-n-2 = -n
h(1) = (m+7)(-n) = -40
Do the same for h(-1) and then solve for n and m. Check your final answer.