ACT Math : Algebraic Functions
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Example Question #2792 : Sat Mathematics
A function F is defined as follows:
for x2 > 1, F(x) = 4x2 + 2x – 2
for x2 < 1, F(x) = 4x2 – 2x + 2
What is the value of F(1/2)?
For F(1/2), x2=1/4, which is less than 1, so we use the bottom equation to solve. This gives F(1/2)= 4(1/2)2 – 2(1/2) + 2 = 1 – 1 + 2 = 2
Example Question #11 : Algebraic Functions
Which of the statements describes the solution set for –7(x + 3) = –7x + 20 ?
There are no solutions.
x = –1
All real numbers are solutions.
x = 0
There are no solutions.
By distribution we obtain –7x – 21 = – 7x + 20. This equation is never possibly true.
Example Question #12 : Algebraic Functions
Will just joined a poetry writing group in town that meets once a week. The number of poems Will has written after a certain number of meetings can be represented by the function 
For this function, simply plug 7 in for 
Example Question #13 : Algebraic Functions
If 
To find 
Thus, 
We expand 
We can combine like terms to get 
We add 3 to this result to get our final answer.
Example Question #15 : Algebraic Functions
What is the value of the function f(x) = 6x2 + 16x – 6 when x = –3?
–108
96
–12
0
0
There are two ways to do this problem. The first way just involves plugging in –3 for x and solving 6〖(–3)〗2 + 16(–3) – 6, which equals 54 – 48 – 6 = 0. The second way involves factoring the polynomial to (6x – 2)(x + 3) and then plugging in –3 for x. The second way quickly shows that the answer is 0 due to multiplying by (–3 + 3).
Example Question #14 : Algebraic Functions
Given the functions f(x) = 2x + 4 and g(x) = 3x – 6, what is f(g(x)) when x = 6?
144
12
28
192
16
28
We need to work from the inside to the outside, so g(6) = 3(6) – 6 = 12.
Then f(g(6)) = 2(12) + 4 = 28.
Example Question #15 : Algebraic Functions
A function f(x) = –1 for all values of x. Another function g(x) = 3x for all values of x. What is g(f(x)) when x = 4?
–1
–3
3
–12
12
–3
We work from the inside out, so we start with the function f(x). f(4) = –1. Then we plug that value into g(x), so g(f(x)) = 3 * (–1) = –3.
Example Question #17 : Algebraic Functions
What is f(–3) if f(x) = x2 + 5?
–4
–14
4
15
14
14
f(–3) = (–3)2 + 5 = 9 + 5 = 14
Example Question #21 : Algebraic Functions
For all values of x, f(x) = 7x2 – 3, and for all values of y, g(y) = 2y + 9. What is g(f(x))?
14y2 + 3
2x + 9
14x2 + 3
7y2 – 3
14x2 – 3
14x2 + 3
The inner function f(x) is like our y-value that we plug into g(y).
g(f(x)) = 2(7x2 – 3) + 9 = 14x2 – 6 + 9 = 14x2 + 3.
Example Question #22 : Algebraic Functions
Find
Simply plug 6 into the equation and don't forget the absolute value at the end.
absolute value = 67
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