Algebraic Functions - GRE Quantitative Reasoning

With composition of functions (as with the order of operations) we perform what is inside of the parentheses first. So, g(3)=2(3)+2=8 and then f(8)=24.

Substitute -1 for in the given function.

If you didn’t remember the negative sign, you will have calculated 36. If you remembered the negative sign at the very last step, you will have calculated -36; however, if you did not remember that is 1, then you will have calculated 18.

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).

Plug g(x) into f(x) as if it is just a variable. This gives f(g(x)) = 3(x2 – 12) + 7.

Distribute the 3: 3x2 – 36 + 7 = 3x2 – 29

We can take each of the listed transformations of one at a time. If is to be shifted up by four units, increase every value of by 4.

Next, take this equation and reflect it across the x-axis. If we reflect a function across the x-axis, then all of its values will be multiplied by negative one. So, can be written in the following way:

Lastly, distribute the negative sign to arrive at the final answer.

Adding 12 to both sides and dividing by 4 yields (7y+12)/4.

By distribution we obtain **–**7x – 21 = – 7x + 20. This equation is never possibly true.

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

It is important to follow the order of operations for this equation and find a solution that satisfies both F(x) and G(x).

Recall the order of operations is PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

The correct answer is 4 because

F(x) = 2_x_ + 3_x_ + (9_x_/3) = 2(4) + 34 + ((9 * 4)/3) = 101, and

G(x) = (((24 + 44)/2) - 4 * 4) – 5(4) + 1 = 101.

A composite function substitutes one function into another function and then simplifies the resulting expression. F(G(x)) means the G(x) gets put into F(x).

F(G(x)) = 2(x – 3)2 + 3 = 2(x2 – 6x +9) + 3 = 2x2 – 12x + 18 + 3 = 2x2 – 12x + 21

G(F(x)) = (2x2 +3) – 3 = 2x2

When functions are set up within other functions like in this problem, the function closest to the given variable is performed first. The value obtained from this function is then plugged in as the variable in the outside function. Since b(x) = –2x, and x = 2, the value we obtain from b(x) is –4. We then plug this value in for x in the a(x) function. So a(x) then becomes 2(–43) + (–4), which equals –132.

F(G(x)) is a composite function where the expression G(x) is substituted in for x in F(x)

F(G(x)) = (x + 5)3 + 2(x + 5)2 – 3 = x_3 + 17_x_2 + 95_x + 172

G(F(x)) = _x_3 + _x_2 + 2

F(x) – G(x) = _x_3 + 2_x_2 – x – 8

F(x) + G(x) = _x_3 + 2_x_2 + x + 2

Evaluating yields –6174.

–147(–21 + 63) =

–147 * 42 = –6174

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.

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.

f(–3) = (–3)2 + 5 = 9 + 5 = 14

The inner function f(x) is like our y-value that we plug into g(y).

g(f(x)) = 2(7_x_2 – 3) + 9 = 14_x_2 – 6 + 9 = 14_x_2 + 3.

Simply plug 6 into the equation and don't forget the absolute value at the end.

absolute value = 67