If f (x)=x+2 , g(x)=-4x+3, and h(x)=x^2-2x+1, find each value.
1. (f x g x h) (3)
2. (h divided by fg) (-6)
3. [ g o ( h o f)] (-4)
If f(x)=5x , h(x)=x^2+6x+8 , solve f[h(a+4)]
I think you can easily fill in the first two. Simple multiplication and division, right?
(g◦(h◦f))(-4)
= g(h(f(-4)))
= g(h(-2))
= g(9)
= -33
Note that h(x) = (x-1)^2. So,
f(h(a+4))
= f((a+4-1)^2))
= f((a+3)^2)
= (a+3)^2+2
You can expand that if you want.
To find the value of an expression involving functions, you need to substitute the specified value into each function and evaluate the resulting expression. Let's solve each problem step by step:
1. (f x g x h) (3):
To evaluate this expression, we need to substitute the value 3 into each function and then perform the operations in the correct order.
- First, let's find f(3): f(3) = 3 + 2 = 5.
- Next, we find g(3): g(3) = -4(3) + 3 = -12 + 3 = -9.
- Lastly, we find h(3): h(3) = 3^2 - 2(3) + 1 = 9 - 6 + 1 = 4.
Finally, we substitute these results back into the expression: (f x g x h) (3) = f(g(h(3))) = f(g(4)) = f(-9) = -9 + 2 = -7.
2. (h divided by fg) (-6):
To evaluate this expression, we first find the values of the functions h, f, and g at -6, and then perform the operations in the correct order.
- First, we find h(-6): h(-6) = (-6)^2 - 2(-6) + 1 = 36 + 12 + 1 = 49.
- Next, we find f(-6): f(-6) = 5(-6) = -30.
- Then, we find g(-6): g(-6) = -4(-6) + 3 = 24 + 3 = 27.
Finally, we substitute these results back into the expression: (h divided by fg) (-6) = h(-6) / (f(-6) * g(-6)) = 49 / (-30 * 27) = 49 / (-810) = -49 / 810 (you may want to reduce the fraction further if needed).
3. [g o (h o f)] (-4):
In this problem, we need to first evaluate the innermost function and then substitute it into the outer function.
- First, let's find h o f: h o f = h(f(x)) = h(5x) = (5x)^2 - 2(5x) + 1 = 25x^2 - 10x + 1.
- Next, we substitute this result into g: g o (h o f) = g(h o f) = g(25x^2 - 10x + 1).
- Finally, we evaluate g(25x^2 - 10x + 1) by substituting -4 into the expression: g(25(-4)^2 - 10(-4) + 1) = g(400 + 40 + 1) = g(441) = -4(441) + 3 = -1764 + 3 = -1761.
Now let's solve the next problem:
If f(x) = 5x and h(x) = x^2 + 6x + 8, we are asked to find f[h(a+4)]. To solve this, we substitute (a+4) into h and then substitute the result into f(x).
- First, we find h(a + 4): h(a + 4) = (a + 4)^2 + 6(a + 4) + 8 = a^2 + 8a + 16 + 6a + 24 + 8 = a^2 + 14a + 48.
- Next, we substitute this result into f(x): f[h(a + 4)] = f(a^2 + 14a + 48) = 5(a^2 + 14a + 48) = 5a^2 + 70a + 240.
Therefore, f[h(a+4)] = 5a^2 + 70a + 240.