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High School Math : Understanding Functional Notations

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Example Questions

Example Question #1 : Functions And Graphs

Which analysis can be performed to determine if an equation is a function?

Possible Answers:

Horizontal line test

Calculating zeroes

Vertical line test

Calculating domain and range

Correct answer:

Vertical line test

Explanation:

The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one $\small y$ (or $\small f(x)$ ) value for each value of $\small x$ . The vertical line test determines how many $\small y$ (or $\small f(x)$ ) values are present for each value of $\small x$ . If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.

The horizontal line test can be used to determine if a function is one-to-one, that is, if only one $\small x$ value exists for each $\small y$ (or $\small f(x)$ ) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.

Example of a function: y=x^2-6x

Example of an equation that is not a function: y^2-2x+y=7

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Example Question #1 : Function Notation

Let f(x)=2-2x+x^2 and g(x)=2x-1 . What is f(g(x)) ?

Possible Answers:

x^2+3

4x^2-4x+5

2x^2-4x+3

2x^3-5x^2+6x-2

4x^2-8x+5

Correct answer:

4x^2-8x+5

Explanation:

THe notation f(g(x)) is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).

The original expression for f(x) is 2-2x+x^2 . We will take each x and substitute in the value of g(x), which is 2x-1.

f(g(x))=f(2x-1)

=2-2(2x-1)+(2x-1)^2

We will now distribute the -2 to the 2x - 1.

2-4x+2+(2x-1)^2

We must FOIL the (2x-1)^2 term, because (2x-1)^2=(2x-1)(2x-1) .

2-4x+2+(2x-1)^2=2-4x+2+(2x-1)(2x-1)

=2-4x+2+4x^2-2x-2x+1

Now we collect like terms. Combine the terms with just an x.

2+2+4x^2+1-8x

Combine constants.

4x^2-8x+5

The answer is 4x^2-8x+5 .

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Example Question #11 : Functions And Graphs

If $f(x)=3x^{2}+5$ and g(x)= x-2 , what is g(f(x)) ?

Possible Answers:

$3x^{2}+3$

$2x^{2}+5$

$3x^{2}+12x-7$

$3x^{2}-12x+17$

$x^{2}-2$

Correct answer:

$3x^{2}+3$

Explanation:

g(f(x)) means f(x) gets plugged into g(x) .

Thus $g(f(x))= (3x^{2}+5) - 2 = 3x^{2}+3$ .

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Example Question #3 : Understanding Functional Notations

Let $f(x)= x^{3}-2x^{2}+x$ and $g(x)=x^{2}-1$ . What is f(g(2)) ?

Possible Answers:

Correct answer:

Explanation:

Calculate g(2) and plug it into f(x) .

$g(2)= (2)^{2}-1=4-1=3$

$f(3)=(3)^{3}-2(3)^{2}+(3)=27-2(9)+(3)=27-18+3=12$

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Example Question #2 : Function Notation

Evaluate $\small f(g(3))$ if f(x)=6x-4 and g(x)=x^2 .

Possible Answers:

Undefined

196

Correct answer:

Explanation:

$\small f(g(3))$

This expression is the same as saying "take the answer of $\small g(3)$ and plug it into $\small f(x)$ ."

First, we need to find $\small g(3)$ . We do this by plugging $\small 3$ in for $\small x$ in $\small g(x)$ .

g(x)=x^2

g(3)=3^2=9

Now we take this answer and plug it into $\small f(x)$ .

f(g(3))=f(9)

We can find the value of $\small f(9)$ by replacing $\small x$ with $\small 9$ .

f(x)=6x-4

f(9)=6(9)-4=50

This is our final answer.

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High School Math : Understanding Functional Notations

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Functions And Graphs

Example Question #1 : Function Notation

Example Question #11 : Functions And Graphs

Example Question #3 : Understanding Functional Notations

Example Question #2 : Function Notation

All High School Math Resources

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