### All ACT Math Resources

## Example Questions

### Example Question #11 : Algebraic Functions

If f(x)=3x and g(x)=2x+2, what is the value of f(g(x)) when x=3?

**Possible Answers:**

18

20

24

22

**Correct answer:**

24

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.

### Example Question #1 : How To Find F(X)

g(x) = 4x – 3

h(x) = .25πx + 5

If f(x)=g(h(x)). What is f(1)?

**Possible Answers:**

π + 17

19π – 3

42

4

13π + 3

**Correct answer:**

π + 17

First, input the function of h into g. So f(x) = 4(.25πx + 5) – 3, then simplify this expression f(x) = πx + 20** –** 3 (leave in terms of π** **since our answers are in terms of π). Then plug in 1 for x to get π** **+ 17.

### Example Question #1 : How To Find F(X)

If 7y = 4x - 12, then x =

**Possible Answers:**

**Correct answer:**(7y+12)/4

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

### Example Question #1 : How To Find F(X)

What is ?

**Possible Answers:**

**Correct answer:**

### Example Question #2 : How To Find F(X)

If F(x) = 2x^{2} + 3 and G(x) = x – 3, what is F(G(x))?

**Possible Answers:**

2x^{2}

2x^{2} + 12x +18

2x^{2} – 12x +21

6x^{2} + 5x

6x^{2} – 12x

**Correct answer:**

2x^{2} – 12x +21

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(x^{2} – 6x +9) + 3 = 2x^{2} – 12x + 18 + 3 = 2x^{2} – 12x + 21

G(F(x)) = (2x^{2} +3) – 3 = 2x^{2}

### Example Question #2 : How To Find F(X)

If a(x) = 2x^{3 }+ x, and b(x) = –2x, what is a(b(2))?

**Possible Answers:**

132

128

–132

503

–503

**Correct answer:**

–132

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(–4^{3}) + (–4), which equals –132.

### Example Question #2 : Algebraic Functions

Let *F*(*x*) = *x*^{3} + 2*x*^{2} – 3 and G(*x*) = *x* + 5. Find *F*(*G*(*x*))

**Possible Answers:**

*x*^{3} + *x*^{2} + 2

*x*^{3} + *x*^{2} + *x* + 8

*x*^{3} + 17*x*^{2} + 95*x* + 172

*x*^{3} + 2*x*^{2} – *x* – 8

*x*^{3} + 2*x*^{2} + *x* + 2

**Correct answer:**

*x*^{3} + 17*x*^{2} + 95*x* + 172

*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

### Example Question #3 : Algebraic Functions

What is the value of *xy*^{2}(*xy –* 3*xy*) given that *x *= –3 and *y *= 7?

**Possible Answers:**

–6174

2881

–2881

3565

**Correct answer:**

–6174

Evaluating yields –6174.

–147(–21 + 63) =

–147 * 42 = –6174

### Example Question #2 : How To Find F(X)

Find .

**Possible Answers:**

**Correct answer:**

is . To start, we find that . Using this, we find that .

Alternatively, we can find that . Then, we find that .

### Example Question #1 : Algebraic Functions

It takes no more than 40 minutes to run a race, but at least 30 minutes. What equation will model this in m minutes?

**Possible Answers:**

**Correct answer:**

If we take the mean number of minutes to be 35, then we need an equation which is less than 5 from either side of 35. If we subtract 35 from minutes and take the absolute value, this will give us our equation since we know that the time it takes to run the marathon is between 30 and 40 minutes.