Introduction to Functions

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Algebra II › Introduction to Functions

Questions 1 - 10

1

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: $(-\infty, 4]$

2

Determine the inverse of:

$y=\frac{1}{7}x+\frac{4}{7}$

$y=\frac{1}{7}x-\frac{4}{7}$

$y=-7x+\frac{4}{7}$

$y=\frac{4}{7}x-\frac{4}{7}$

Explanation

Interchange the x and y variables and solve for y.

Add four on both sides.

Simplify both sides.

Divide by seven on both sides.

$\frac{x+4}{7}= \frac{7y}{7}$

The answer is: $y=\frac{1}{7}x+\frac{4}{7}$

3

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:

4

Determine the inverse of: $y=\frac{2}{5}x-\frac{1}{2}$

$y=\frac{5}{2}x+\frac{5}{4}$

$y=\frac{5}{2}x+\frac{1}{2}$

$y=-\frac{5}{2}x-\frac{1}{2}$

$y=-\frac{5}{2}x+\frac{1}{2}$

$y=\frac{5}{2}x-\frac{5}{4}$

Explanation

Interchange the x and y-variables.

$x=\frac{2}{5}y-\frac{1}{2}$

Solve for y. Add one-half on both sides.

$x+\frac{1}{2}=\frac{2}{5}y-\frac{1}{2}+\frac{1}{2}$

Simplify both sides.

$x+\frac{1}{2}=\frac{2}{5}y$

Multiply five over two on both sides in order to isolate the y-variable.

$\frac{5}{2}(x+\frac{1}{2})=\frac{2}{5}y\cdot \frac{5}{2}$

Apply the distributive property on the left side. The right side will reduce to just a lone y-variable.

The answer is: $y=\frac{5}{2}x+\frac{5}{4}$

5

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

6

What is the domain and range of the following graph?

Graph for questions

Domain: All real numbers

Range: $y\leq9$

Domain: $y\leq9$

Range: All real numbers

Domain: All real numbers

Range: $x\leq9$

Domain: $x\leq9$

Range: All real numbers

Domain: All real numbers

Range: All real numbers

Explanation

Domain looks at x-values and range looks at y-values.

The x-values appear to continue to go on forever, which suggests the answer:

"all real numbers"

The y-values are all number that are equal to nine or less which is

$y\leq9$

So you answer is:

Domain: All real numbers

Range: $y\leq9$

7

$\textup{What is the domain of y = x ?}$

Explanation

All inputs are valid. There is nothing you can put in for x that won't work.

8

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:

9

Determine the inverse of: $y=\frac{2}{5}x-\frac{1}{2}$

$y=\frac{5}{2}x+\frac{5}{4}$

$y=\frac{5}{2}x+\frac{1}{2}$

$y=-\frac{5}{2}x-\frac{1}{2}$

$y=-\frac{5}{2}x+\frac{1}{2}$

$y=\frac{5}{2}x-\frac{5}{4}$

Explanation

Interchange the x and y-variables.

$x=\frac{2}{5}y-\frac{1}{2}$

Solve for y. Add one-half on both sides.

$x+\frac{1}{2}=\frac{2}{5}y-\frac{1}{2}+\frac{1}{2}$

Simplify both sides.

$x+\frac{1}{2}=\frac{2}{5}y$

Multiply five over two on both sides in order to isolate the y-variable.

$\frac{5}{2}(x+\frac{1}{2})=\frac{2}{5}y\cdot \frac{5}{2}$

Apply the distributive property on the left side. The right side will reduce to just a lone y-variable.

The answer is: $y=\frac{5}{2}x+\frac{5}{4}$

10

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

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