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PSAT Math : Algebraic Functions

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Example Questions

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Example Question #61 : How To Find F(X)

If $f(x)=x+\frac{1}{x}$ , find the value of f(4) .

Possible Answers:

$\frac{41}{4}$

$\frac{17}{4}$

Correct answer:

$\frac{17}{4}$

Explanation:

f(4) = 4 + 1/4 = 16/4 + 1/4 = 17/4

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Example Question #321 : Algebra

If f(x) = x + 8 and g(x) = x^2 + x , what is g(f(3)) ?

Possible Answers:

104

132

100

Correct answer:

132

Explanation:

g is a function of f, and f is a function of 3, so you must work inside out.

f(3) = 11

g(f(3)) = g(11) = 121 + 11 = 132

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Example Question #63 : How To Find F(X)

Jamie buys a new fish tank. It is 45% full. She adds seven more liters of water, and it is 65% full. What is the capacity of Jamie's new fish tank?

Possible Answers:

35 liters

None of the other answers are correct.

48 liters

22 liters

27 liters

Correct answer:

35 liters

Explanation:

The algebraic expression for x, the capacity of the fishtank is:

$0.45x+7=0.65x$

$0.45x+7-0.45x=0.65x-0.45x$

$0.2x=7$

$0.2x\times\frac{1}{0.2}=7\times\frac{1}{0.2}$

$x=35\hspace{1 mm}L$

The capacity of the fishtank is 35 liters.

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Example Question #1031 : Algebra

f(x)=2x+6

g(x)=3x-3

2f(x)-g(x)+2=?

Possible Answers:

x + 17

7x + 17

x + 11

5x + 11

x + 5

Correct answer:

x + 17

Explanation:

When we multiply a function by a constant, we multiply each value in the function by that constant. Thus, 2f(x) = 4x + 12. We then subtract g(x) from that function, making sure to distribute the negative sign throughout the function. Subtracting g(x) from 4x + 12 gives us 4x + 12 - (3x - 3) = 4x + 12 - 3x + 3 = x + 15. We then add 2 to x + 15, giving us our answer of x + 17.

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Example Question #64 : How To Find F(X)

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 #31 : Algebraic Functions

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

Possible Answers:

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

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

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

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

$x^{2}+3+h$

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 #101 : 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:

$\frac{4}{x}$

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

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

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

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

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 #108 : Algebraic Functions

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

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

Possible Answers:

$\small 8$

$\small 10$

$\small 2$

$\small 14$

$\small 6$

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 #65 : How To Find F(X)

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 #61 : How To Find F(X)

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

Possible Answers:

f(1)

f(-2)

f(-1)

f(0)

f(-4)

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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PSAT Math : Algebraic Functions

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #61 : How To Find F(X)

Example Question #321 : Algebra

Example Question #63 : How To Find F(X)

Example Question #1031 : Algebra

Example Question #64 : How To Find F(X)

Example Question #31 : Algebraic Functions

Example Question #101 : Algebraic Functions

Example Question #108 : Algebraic Functions

Example Question #65 : How To Find F(X)

Example Question #61 : How To Find F(X)

All PSAT Math Resources

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