### All PSAT Math Resources

## Example Questions

### Example Question #1 : Algebraic Functions

If *f*(*x*) = *x*^{2} – 5 for all values *x* and *f*(*a*) = 4, what is one possible value of *a*?

**Possible Answers:**

3

1

8

11

14

**Correct answer:**

3

Using the defined function, *f*(*a*) will produce the same result when substituted for *x*:

*f*(*a*) = *a*^{2} – 5

Setting this equal to 4, you can solve for *a*:

*a*^{2} – 5 = 4

*a*^{2} = 9

*a* = –3 or 3

### Example Question #2 : Algebraic Functions

If the function *g* is defined by *g*(*x*) = 4*x* + 5, then 2*g*(*x*) – 3 =

**Possible Answers:**

4*x* + 2

6*x* + 7

8*x* + 2

6*x* + 2

8*x* + 7

**Correct answer:**

8*x* + 7

The function *g*(*x*) is equal to 4*x* + 5, and the notation 2*g*(*x*) asks us to multiply the entire function by 2. 2(4*x* + 5) = 8*x* + 10. We then subtract 3, the second part of the new equation, to get 8*x* + 7.

### Example Question #3 : Algebraic Functions

If *f*(*x*) = *x*^{2} + 5*x* and *g*(*x*) = 2, what is *f*(*g*(4))?

**Possible Answers:**

36

14

4

2

39

**Correct answer:**

14

First you must find what *g*(4) is. The definition of *g*(*x*) tells you that the function is always equal to 2, regardless of what “*x*” is. Plugging 2 into *f*(*x*), we get 2^{2} + 5(2) = 14.

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

*f*(*a*) = ^{1}/_{3}(*a*^{3} + 5*a* – 15)

Find *a *= 3.

**Possible Answers:**

27

19

3

1

9

**Correct answer:**

9

Substitute 3 for all a.

(1/3) * (3^{3} + 5(3) – 15)

(1/3) * (27 + 15 – 15)

(1/3) * (27) = 9

### Example Question #5 : Algebraic Functions

Evaluate *f*(*g*(6)) given that *f*(*x*) = *x*^{2} – 6 and *g*(*x*) = –(1/2)*x* – 5

**Possible Answers:**

50

–8

30

–25

58

**Correct answer:**

58

Begin by solving *g*(6) first.

*g*(6) = –(1/2)(6) – 5

*g*(6) = –3 – 5

*g*(6) = –8

We substitute *f*(–8)

*f*(–8) = (–8)^{2} – 6

*f*(–8) = 64 – 6

*f*(–8) = 58

### Example Question #6 : Algebraic Functions

If *f*(*x*) = |(*x*^{2 }– 175)|, what is the value of *f*(–10) ?

**Possible Answers:**

–275

75

15

275

–75

**Correct answer:**

75

If *x* = –10, then (*x*^{2} – 175) = 100 – 175 = –75. But the sign |*x*| means the absolute value of *x*. Absolute values are always positive.

|–75| = 75

### Example Question #7 : Algebraic Functions

If f(x)= 2x² + 5x – 3, then what is f(–2)?

**Possible Answers:**

–5

–21

–1

7

**Correct answer:**

–5

By plugging in –2 for x and evaluating, the answer becomes 8 – 10 – 3 = -5.

### Example Question #8 : Algebraic Functions

If f(x) = x² – 2 and g(x) = 3x + 5, what is f(g(x))?

**Possible Answers:**

9x² + 30x + 25

9x² + 30x + 23

3x² – 1

9x² + 23

**Correct answer:**

9x² + 30x + 23

To find f(g(x) plug the equation for g(x) into equation f(x) in place of “x” so that you have: f(g(x)) = (3x + 5)² – 2.

Simplify: (3x + 5)(3x + 5) – 2

Use FOIL: 9x² + 30x + 25 – 2 **= **9x² + 30x + 23

### Example Question #9 : Algebraic Functions

f(x) = 2x^{2} + x – 3 and g(y) = 2y – 7. What is f(g(4))?

**Possible Answers:**

57

0

33

-33

42

**Correct answer:**

0

To evaluate f(g(4)), one must first determine the value of g(4), then plug that into f(x).

g(4) = 2 x 4 – 7 = 1.

f(1) = 2 x 1^{2} + 2 x 1 – 3 = 0.

### Example Question #10 : Algebraic Functions

For all positive integers, let k***** be defined by k*** = **(k-1)(k+2) . Which of the following is equal to 3*****+4*****?

**Possible Answers:**

5*

7*

6*

4*

**Correct answer:**

5*

We can think of k❋ as the function f(k)=(k-1)(k+2), so 3❋+4❋is f(3)+f(4). When we plug 3 into the function, we find f(3)=(3-1)(3+2)=(2)(5)=10, and when we plug 4 into the function, we find f(4)=(4-1)(4+2)=(3)(6)=18, so f(3)+f(4)=10+18=28. The only answer choice that equals 28 is 5❋ which is f(5)=(5-1)(5+2)=(4)(7)=28.

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