ACT study guide Math: Functions

Let x⊳y=y/x for real numbers x and y. For what values of x is the value of x⊳(−1) always positive?

(If x⊳(−1)=−1x is positive, x must be negative, since a negative number divided by a negative number is always positive.) x < 0

Let x△y=x+1/y for integers x and y. What is the value of 7△(−2)?

7△(−2)=7+1/−2=8−2= −4

If f(a) = 3a2 − a3, then what is the value of f(−2)?

f(−2) = 3(−2)2 − (−2)3 = 3(4) + 8 = 20

Which operation in place of the △ in the following equation would make the equation true for all nonzero values of x? x^2△2x=x/2

(Since the term on the right side of the equation has a lower degree than the first term on the left side of the equation, the operation must have been) division

The point (q, r) lies on the graph of a function f(x) in the (x, y) coordinate plane. If the graph of g(x) results from shifting the graph of f(x) up 10 units, then which of the following points must lie on the graph of g(x)?

(The graph of the function f(x) moved up 10 units will be represented by the function f(x) + 10. This means every y-coordinate will be increased by 10 units.) q, r + 10

Let a Ξ b = b2 − a2 + 1 for any real numbers a and b. What is the value of (−2) Ξ (−3)?

(−2) Ξ (−3) = (−3)2 −(−2)2 + 1 = 9 − 4 + 1 = 6

If f (x) = 2x2 + 3, then f (x + h) = ?

(To find f (x + h) when f (x) = 2x2 + 3, substitute (x + h) for x in f (x) = 2x2 + 3, and expand the equation as follows: f (x + h) = 2(x + h)2 + 3 f (x + h) = 2(x2 + 2xh + h2) + 3) f (x + h) = 2x2 + 4xh + 2h2 + 3

The length L, in meters, of a spring is given by the equation L=(23)F+0.05, where F is the applied force in newtons. Approximately what force, in newtons, must be applied for the spring's length to be 0.23 meters

(To find the force F (in newtons) corresponding to the spring length, L, of 0.23 meters when the relationship is given by the equation L=(23)F+0.05, first substitute 0.23 for L to get 0.23=(23)F+0.05. Next, subtract 0.05 from both sides to get 0.18=(23)F. Finally, multiply by (32), since dividing by a fraction is equal to multiplying by its reciprocal, to arrive at the answer for F which is )0.27

The table below gives values of the quadratic function f for selected values of x. Which of the following defines the quadratic function f? x 0 1 2 3 f(x) −6 −5 −2 3

(To solve this problem, substitute the given values for x into each function to see if you get the correct value for f (x). Answer choice A is correct because for each of the four given x values, x2 − 6 yields the correct f (x) values) f(x) = x2 − 6

The value of an investment grows based on the function V(t) = 1000(1 + 0.06)t, where V is the value in dollars and t is the time in years. To the nearest cent, what is the value of the investment after 8 years?

(V(8) = 1000(1 + 0.06)8 =) 1593.85

Let x ◙ y = (x -2y)2 for all integers x and y. Which of the following is the value of 5 ◙ (-3)?

(You are given that x ◙ y = (x − 2y)2 and are asked to solve 5 ◙ (−3). To do this, simply replace x with 5, and y with −3, as follows: x ◙ y = (x − 2y)2 5 ◙ (−3) = (5 − 2(−3))2 5 ◙ (−3) = (5 − (−6))2 5 ◙ (−3) = (5 + 6)2 5 ◙ (−3) = (11)2 5 ◙ (−3) =) 121

Define a ∘ b = 2a − 3b for all integers a and b. What is the value of −2 ∘ 1?

(−2∘1= 2(−2)−3(1)=−4−3 =)−7

If f (x) = −3x2 − 8, then f (−4) = ?

substitute −4 for x in the expression −3x2 − 8, as follows (remember to keep track of the negative signs!): −3(−4)2 − 8 = −3(16) − 8 −48 − 8 = −56 If you did not keep track of the negative signs, you would most likely arrive at one of the incorrect answer choices