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PSAT Math : Algebra

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Example Questions

Example Question #986 : Psat Mathematics

If $f(t)=6+(4-t)^2$ , what is the smallest possible value of $f(t)$ ?

Possible Answers:

Correct answer:

Explanation:

This equation describes a parabola whose vertex is located at the point (4, 6). No matter how large or small the value of t gets, the smallest that f(t) can ever be is 6 because the parabola is concave up. To prove this to yourself you can plug in different values of t and see if you ever get anything smaller than 6.

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Example Question #61 : Algebraic Functions

If $f(x)=x^{2}+3$ , then $f(x+h)=$ ?

Possible Answers:

$x^{2}+2xh+h^{2}+3$

$x^{2}+2xh+h^{2}$

$x^{2}+h^{2}$

$x^{2}+h^{2}+3$

$x^{2}+3+h$

Correct answer:

$x^{2}+2xh+h^{2}+3$

Explanation:

To find $f(x+h)$ when $f(x)=x^{2}+3$ , we substitute $(x+h)$ for $x$ in $f(x)$ .

Thus, $f(x+h)=(x+h)^{2}+3$ .

We expand $(x+h)^{2}$ to $x^{2}+xh+xh+h^{2}$ .

We can combine like terms to get $x^{2}+2xh+h^{2}$ .

We add 3 to this result to get our final answer.

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Example Question #62 : Algebraic Functions

Let $f(x)$ and $g(x)$ be functions such that $f(x)=\frac{1}{x-3}$ , and $f(g(x))=g(f(x))=x$ . Which of the following is equal to $g(x)$ ?

Possible Answers:

$\frac{x}{x-3}$

$\frac{3x+1}{x}$

$\frac{x}{x+3}$

$\frac{1}{x+3}$

$\frac{4}{x}$

Correct answer:

$\frac{3x+1}{x}$

Explanation:

If $\dpi{100} h(x)$ and $\dpi{100} k(x)$ are defined as inverse functions, then $\dpi{100} h(k(x))=k(h(x))=x$ . Thus, according to the definition of inverse functions, $\dpi{100} f(x)$ and $\dpi{100} g(x)$ given in the problem must be inverse functions.

If we want to find the inverse of a function, the most straighforward method is usually replacing $\dpi{100} f(x)$ with $\dpi{100} y$ , swapping $\dpi{100} y$ and $\dpi{100} x$ , and then solving for $\dpi{100} y$ .

We want to find the inverse of $f(x)=\frac{1}{x-3}$ . First, we will replace $\dpi{100} f(x)$ with $\dpi{100} y$ .

$y = \frac{1}{x-3}$

Next, we will swap $\dpi{100} x$ and $\dpi{100} y$ .

$x = \frac{1}{y-3}$

Lastly, we will solve for $\dpi{100} y$ . The equation that we obtain in terms of $\dpi{100} x$ will be in the inverse of $\dpi{100} f(x)$ , which equals $\dpi{100} g(x)$ .

We can treat $x = \frac{1}{y-3}$ as a proportion, $\frac{x}{1} = \frac{1}{y-3}$ . This allows us to cross multiply and set the results equal to one another.

$x(y-3)= 1$

We want to get y by itself, so let's divide both sides by x.

$y-3=\frac{1}{x}$

Next, we will add 3 to both sides.

$y=\frac{1}{x}+3$

To combine the right side, we will need to rewrite 3 so that it has a denominator of $\dpi{100} x$ .

$y=\frac{1}{x}+3 = \frac{1}{x}+\frac{3x}{x}=\frac{3x+1}{x}$

The answer is $\frac{3x+1}{x}$ .

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Example Question #987 : Psat Mathematics

$\small \textup{The function}\, f(x)\, \textup{is defined as }f\left ( x\right )=5x-2$ .

$\small \small f\left ( 4\right )-f\left ( 2\right )=\,?$

Possible Answers:

$\small 14$

$\small 8$

$\small 2$

$\small 6$

$\small 10$

Correct answer:

$\small 10$

Explanation:

$\small f\left ( x\right )=5x - 2$

$\small f\left ( 4\right )=5\left ( 4\right )-2 = 18$

$\small f\left ( 2\right )=5\left ( 2\right )-2 = 8$

$\small 18-8 = 10$

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Example Question #989 : Psat Mathematics

Let the function f be defined by f(x)=x-t. If f(12)=4, what is the value of f(0.5*t)?

Possible Answers:

Correct answer:

Explanation:

First we substitute in 12 for x and set the equation up as 12-t=4. We then get t=8, and substitute that for t and get f(0.5*8), giving us f(4). Plugging 4 in for x, and using t=8 that we found before, gives us:

f(4) = 4 - 8 = -4

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Example Question #63 : Algebraic Functions

Which of the following is equal to if $f(x) = (x+3)^{2}-3$ ?

Possible Answers:

f(-4)

f(0)

f(-1)

f(-2)

f(1)

Correct answer:

f(0)

Explanation:

To solve, we can set the given function equal to six and solve for .

$6= (x+3)^{2}-3$

Add 3 to each side:

9=(x+3)^2

Take the square root. Remember that the root can be positive OR negative:

$\pm3=x+3$

Subtract 3 from each side. It will be easiest to separate the equation into two parts:

$3=x+3\ \text{or}\ -3=x+3$

$0=x\ \text{or}\ -6=x$

Now we know that is equal to or . Based on the available answer options, the correct choice must be f(0) .

You can also solve this question by checking each answer option separately; you should find the same final answer.

Substitute each of the answer choices to see which one makes the equation equal to 6.

$f(1)=(1+3)^{2}-3=4^2-3=16-3=13$

f(-1) = (-1+3)^2-3 = 2^2-3 = 4-3 =1

f(-2) =(-2+3)^2-3 = 1^2 - 3 = 1-3= -2

f(-4)= (-4+3)^2 - 3 = (-1)^2 - 3 = 1-3 = - 2

f(0) = (0+3)^2 -3=9-3=6

The answer to this question is f(0) . No other answer makes the equation equal 6.

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Example Question #331 : Algebra

If is an odd integer, and is an even integer, which of the following must be even?

Possible Answers:

4a+3b

2b-3a

3a-b

ab+1

3a+3b

Correct answer:

4a+3b

Explanation:

Solve this problem using picked numbers. Choose an odd number to represent and an even number to represent . In this case, we have chosen 3 to represent and 2 to represent .

Substitute into each equation to find the correct answer:

4a+3b = 4(3)+3(2)=12+6=18

This number is even, and likely the answer we are looking for. Just in case, quickly check the other answers to make sure no others come out even.

3a+3b = 3(3)+3(2)=9+6=15

This expression gives an odd answer and must be incorrect.

3a-b = 3(3)-2=9-2=7

This expression gives an odd answer and must be incorrect.

2b-3a = 2(2)-3(3)=4-9=-5

This expression gives an odd answer and must be incorrect.

ab+1 = 2*3+1=6+1=7

This expression gives an odd answer and must be incorrect.

Only the first answer is even. The answer is 4a+3b .

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Example Question #72 : How To Find F(X)

If $f\left ( x\right ) = x^{2}+2\left(\frac{x}{4}\right)+\sqrt{x}$

What is f(4) ?

Possible Answers:

$20+\sqrt{4}$

200

Correct answer:

Explanation:

To find f(4), input a 4 in every place you see an x in the equation. That gives you

$f\left ( 4\right ) = 4^{2}+2\left(\frac{4}{4}\right)+\sqrt{4}$

When you simplify this expression, you get

$f\left ( 4\right ) = 16+2(1)+2$

When you add together each part, you get

$f\left ( 4\right ) = 20$

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Example Question #332 : Algebra

What is the value of the function f(x) = 6x²+ 16x – 6 when x = –3?

Possible Answers:

–108

–12

Correct answer:

Explanation:

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).

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Example Question #35 : Algebraic Functions

Given the functions f(x) = 2x + 4 and g(x) = 3x – 6, what is f(g(x)) when x = 6?

Possible Answers:

192

144

Correct answer:

Explanation:

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.

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PSAT Math : Algebra

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #986 : Psat Mathematics

Example Question #61 : Algebraic Functions

Example Question #62 : Algebraic Functions

Example Question #987 : Psat Mathematics

Example Question #989 : Psat Mathematics

Example Question #63 : Algebraic Functions

Example Question #331 : Algebra

Example Question #72 : How To Find F(X)

Example Question #332 : Algebra

Example Question #35 : Algebraic Functions

All PSAT Math Resources

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