Introduction to Functions

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Algebra II › Introduction to Functions

Questions 1 - 10

Find the range of the function:

$(-\infty, 4]$

$(-\infty, 4)$

$(-\infty, -4]$

$(-\infty, 4)\bigcup(4,\infty)$

$(-\infty,\infty )$

Explanation

The range is the existing y-values that contains the function.

Notice that this is a parabola that opens downward, and the y-intercept is four.

This means that the highest y-value on this graph is four. The y-values will approach negative infinity as the domain, or x-values, approaches to positive and negative infinity.

$x\leq4$

The answer is: $(-\infty, 4]$

Find the range of the following equation:

$[0,\infty)$

$[1,\infty)$

$[4,\infty)$

$(-\infty,4]$

$(-\infty,0]$

Explanation

Expand the quadratic.

Use the FOIL method to expand the binomials.

The equation becomes:

Now that we have the equation in format, find the vertex. This will determine the minimum of the parabola.

The formula is:

$x=-\frac{b}{2a}$

Substitute the values.

$x=-\frac{-8}{2(4)} = 1$

To find the y-value, substitute the x-value back to the original equation.

The minimum is:

Because the value of is positive, the parabola will open up.

The range is: $[0,\infty)$

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The above table shows a function with domain $\left \{ -3, -2, -1, 0, 1, 2, 3 \right \}$ .

True or false: has an inverse function.

False

True

Explanation

A function has an inverse function if and only if, for all in the domain of , if $a \ne b$ , it follows that $f(a) \ne f(b)$ . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

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and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

What is the domain of the function $f(x)=\sqrt{7-x}+2$ ?

$(-\infty, 7]$

$[7,\infty)$

$(-\infty, 2]$

$[-7,\infty)$ $[3,\infty)$

$[2,\infty)$

Explanation

The expression under the square root symbol cannot be negative, so to find the domain, set that expression $\geq0$ .

$7-x\geq0$

$-x\geq-2$

$x\leq7$

The domain includes all x-values less than or equal to 7, which can be written as $(-\infty,7]$ .

Determine the inverse:

$y=\frac{1}{10}x-\frac{3}{10}$

$y=\frac{1}{10}x+\frac{3}{10}$

$y=-\frac{1}{10}x-\frac{3}{10}$

$y=\frac{1}{10}x-3$

$y=\frac{3}{10}x-3$

Explanation

In order to find the inverse of this function, interchange the x and y-variables.

Subtract three from both sides.

Simplify the equation.

Divide by ten on both sides.

$\frac{x-3}{10}= \frac{10y}{10}$

Simplify both sides.

The answer is: $y=\frac{1}{10}x-\frac{3}{10}$

If and , determine:

Explanation

Substitute the assigned values into the expression.

Simplify the inside parentheses.

The answer is:

If and , what is $g(f(x))\cdot f(x)$ ?

$\textup{The answer is not given.}$

Explanation

Evaluate first. Substitute the function into .

Distribute the integer through the binomial and simplify the equation.

Multiply this expression with .

The answer is:

Find the range of the following equation:

$[0,\infty)$

$[1,\infty)$

$[4,\infty)$

$(-\infty,4]$

$(-\infty,0]$

Explanation

Expand the quadratic.

Use the FOIL method to expand the binomials.

The equation becomes:

Now that we have the equation in format, find the vertex. This will determine the minimum of the parabola.

The formula is:

$x=-\frac{b}{2a}$

Substitute the values.

$x=-\frac{-8}{2(4)} = 1$

To find the y-value, substitute the x-value back to the original equation.

The minimum is:

Because the value of is positive, the parabola will open up.

The range is: $[0,\infty)$

Find the range of the function:

$(-\infty, 4]$

$(-\infty, 4)$

$(-\infty, -4]$

$(-\infty, 4)\bigcup(4,\infty)$

$(-\infty,\infty )$

Explanation

The range is the existing y-values that contains the function.

Notice that this is a parabola that opens downward, and the y-intercept is four.

This means that the highest y-value on this graph is four. The y-values will approach negative infinity as the domain, or x-values, approaches to positive and negative infinity.

$x\leq4$

The answer is: $(-\infty, 4]$

Which of the following is NOT a function?

$x=\left | y\right |$

$y = x^{3} -x +5$

$y = \sqrt{x-9} +3$

$y = -\sqrt{x^{2}-16}$

Explanation

A function has to pass the vertical line test, which means that a vertical line can only cross the function one time. To put it another way, for any given value of , there can only be one value of . For the function $x=\left | y\right |$ , there is one value for two possible values. For instance, if , then . But if , as well. This function fails the vertical line test. The other functions listed are a line,, the top half of a right facing parabola, $y = \sqrt{x-9} +3$ , a cubic equation, $y = x^{3} -x +5$ , and a semicircle, $y = -\sqrt{x^{2}-16}$ . These will all pass the vertical line test.