ACT study guide Math: composite Functions

If a, b and c are positive integers such that a^b = m and c^2b =n, then mn = ?

If a^b = m

If, for a nonzero value of k, f(x) = x-3/k and f(2) = 2, then k =

If f(2) = 2, then -1/k = 2 and 2k = -1. Therefore, k = -1/2.

given the functions f(x) = x + 6 and g(x) = x^2 -2, which of the following is the value of (f og) (3)

((f og)(3) = f(g(3)) = f(3^2 - 2) = f(7) = 7 + 6 = )13

If f(x) = x + 10 and g(x0 = -5x - 8, then (f og)(7)=

((f og)(7) = f(g(70) = f(-5(7)-8) = f(-43) = -43 + 10 = )-33

Casey has buckets of 3 different sizes. The total capacity of 12 of the buckets is g gallons, the total capacity of 8 buckets of another size is g gallons, and the total capacity of 4 buckets of the third size is also g gallons. In terms of g when g > 0, what is the capacity, in gallons, of each of the smallest-sized buckets?

(First determine the size of each bucket. Because the total capacity of 12 buckets is g gallons, each bucket can hold (1/12)g, or g/12 gallons. Because the total capacity of 8 buckets is g gallons, then each of those buckets can hold (1/8)g, or g/8 gallons. If the total capacity of 4 buckets is g gallons, then each bucket is (1/4)g, or g/4 gallons. Therefore, the capacity of the smallest buckets is ) g/12

The graph of a function g(x) is found by shifting the graph of the function f(x) up by 6 units and t the left by 1 unit. If f(x) = (x - 5)^3, then g(x) =

(In general, if a function f(x) is shifted up by 6 units, the graph will be described by f(x) + 6. If that same graph is shifted to the left one unit, then the graph will be represented by f(x + 1 ) + 6. Applying this to the given expression, (x - 5 + 1) ^3 + 6 = )(x -4)^3

In the (x, y) coordinate plane, the graphs of f(x) = 2x^2 + 14 and g(x) = x^2 + 30 intersect at the point (a, b). If a < 0, what is the value of b?

(Setting the equations equal to each other, 2x^2 + 14 =x^2 + 30. Collecting terms gives us x^2 = 16. This equation has solutions x =4, x = -4. But since a < 0, we will use x = -4. At x = -4, b = f(-4) = 2(16) + 14 = )46

If the point (a, b) lies on the graph of a function f(x) in the standard (x, y ) coordinate plane, then which of the following points lies along the graph of f(x -2)

(The graph of f(x -2) would be the graph of f(x) shifted to the right by 2 units. Therefore, every x-coordinate would be increased by 2) a + 2, b)

For every positive 2-digit number, a, with tens digit x and units digit y, let b be the 2-digit number formed by reversing the digits of a. Which of the following expressions is equivalent to a - b?

(You are given that a is a number with tens digit x and units digit y. Therefore, x is equivalent to 10 times y, and a = xy = 10z + y. You are given that b is formed by reversing the digits of a. Therefore, b = yx =10y + x. Set up an equation and solve for a-b as follows: A-b = (10x + y) - (10y + x) = 10x + y - 10y - x =9x -9y) =9(x -y)

g(x) is a transformation that moves f(x) 2 units in the negative x-direction and 3 units in the positive y-direction in the standard coordinate plane. What is g(x)

(f ( x + 2) shifts the function 2 units to the left. Adding 3 to that shifts g(x) up by 3 units) g(x) = f(x + 2) + 3

If f(x) = x^2 + x - 1, which of the following expressions is equivalent to f(a - 1)

(f (a - 1) = ( a - 1)^2 + (a -1) -1 =a^2 -2a + 1 + a - 1 - 1 =)a^2 - a - 1

Which of the following is equivalent to g(a^2 - 10) when g(x) = x^2 + x - 5

(g(a^2 - 10) = (a^2 - 10)^2 + (a^2 - 10) - 5 = a^4 -20a^2 + 100 +a^2 - 10 -5 =)a^4 - 19a^2 + 85

The length of a side of square is represented as (3x -2) inches. Which of the following general expressions represents the area of the square, in square inches?

(A square with sides (3x - 2) inches would have an area of (3x -2)(3x -2) square inches. Use the"foil" method to multipl all terms in the equation. Adding all of these terms yields the expression 9x^2 - 6x - 6x + 4. After combining like terms, the expression is) 9x^2 - 12x + 4.

If f(x) = x^2 and g(x) = 72 -x, what is the smallest value of x for which f(x) = g(x)?

(If x^2 = 72 - x, then x^2 + x - 72 = 0 Factoring, (x =9)(x-8) = 0, and by the zero product rule, x =-9 and x =8. The question asks for the smaller of these values.) -9

If h(x) = x^1/2 + x^2 and g(x) = 4x - 3, then g(h(4)) =

(h(4) = 4^1/2 + 4^2 = 18 g(h(4)) = g(18) = 4(18) - 3 =) 69

If f(x) = 1/2x^2 and g(x) =2x, then for what value or values of x is f (g(x)) = 8

The equation f (g(x0) = 8 is equivalent to 1/2(2x)^2 = 8, or 1/2 (4x^2) = 8 dividing both sides by 2, x^2 = 4, so x = squareroot of 4 = 2

In the xy-coordinate system, if (r, s) and (r + 2, s + t) are two points on the line defined by the equation y = 4x + 5, then t=?

The slope of the line is equal to 4. In the standard equation for a line, y = mx + b, m is equivalent to the slope. The slope is equal to the change in y- values over the change in x-values; set up the following equation to solve for t: Slope= (s + t) -s/ (r + 2) - r 4 = t/2 8 = t

If f (x) = x^2 + 3, then f (x + y) =

Use the "FOIL" method: (x + y)^2 + 3 (x + y)(x + y) + 3 x^2 + xy + xy + y^2 + 3 x^2 + 2xy + y^2 + 3

If f(x) = x + 2 and g(x) = x^2 - x -6, then which of the following is equivalent to g(x)/f(x) for all values of x not equal to -2

g(x)/f(x) = x^2 - x - 6/ x + 2 = (x-3)(x +2)/ x + 2 = x - 3