Algebraic Functions

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Algebra › Algebraic Functions

Questions 1 - 10

$\small f(x)=2x+17$ .

Find $\small f(-3)$ .

$\small 11$

$\small 23$

$\small 34$

$\small 16$

Explanation

Finding $\small f(-3)$ means plugging in $\small -3$ in place of x into the function $\small f(x)$ . In this case, we are told that $\small f(x)=2x+17$ , so we are multiplying $\small -3$ by 2 and then adding 17:

$\small \small f(-3)=2(-3)+17$ order of opperations tells us to first multiply:

$\small f(-3)=-6+17$ and now add:

$\small f(-3)= 11$

If , then solve

Explanation

We know . When solving a function, we substitute the value of x into the function. In other words, anywhere we see an x, we will replace it with -4.

is a one-to-one function specified in terms of a set of $\left ( x,y \right )$ coordinates:

A = $\left \{ \left ( 0,0 \right ), \left ( 1,2\right ),\left ( 2,4 \right ), \left ( 3,6 \right )\right \}$

Which one of the following represents the inverse of the function specified by set A?

B = $\left \{ \left ( 0,0 \right ), \left ( 0,1 \right ), \left ( 0,2 \right ), \left ( 0,3 \right ) \right \}$

C = $\left \{ \left ( 0,0 \right ), \left ( 2,1 \right ), \left ( 4,2 \right ), \left ( 6,3 \right ) \right \}$

D = $\left \{ \left ( 1,1 \right ), \left ( 2,1 \right ), \left ( -2,1 \right ), \left ( 3,1 \right ) \right \}$

E = $\left \{ \left ( -2,1 \right ), \left ( -2,5 \right ), \left ( -2,3 \right ), \left ( -2, -2 \right ) \right \}$

F = $\left \{ \left ( 2,0 \right ), \left ( 2,-3 \right ),\left ( 2,5 \right ), \left ( 2,-7 \right ) \right \}$

Set C

Set B

Set D

Set E

Set F

Explanation

The set A is an one-to-one function of the form

One can find $f^{-1}\left ( x \right )$ by interchanging the and coordinates in set A resulting in set C.

varies inversely as the square root of . If , then . Find if (nearest tenth, if applicable).

Explanation

The variation equation is $A = \frac{K}{\sqrt{x}}$ for some constant of variation .

Substitute the numbers from the first scenario to find :

$45= \frac{K}{\sqrt{15}}$

$K = 45 \sqrt{15}$

The equation is now $A = \frac{45 \sqrt{15}}{\sqrt{x}}$ .

If , then

$A = \frac{45 \sqrt{15}}{\sqrt{30}} \approx 31.8$

Find the inverse:

$y= \frac{1}{4}x + \frac{3}{2}$

$y= -\frac{1}{4}x + \frac{3}{2}$

$y= \frac{1}{4}x- \frac{3}{2}$

Explanation

To find the inverse, first interchange the x and y variables in the equation.

Solve for y. Add 6 on both sides.

Simplify.

Divide by four on both sides.

$\frac{x+6}{4}=\frac{4y}{4}$

Simplify both sides.

$y=\frac{x+6}{4} = \frac{1}{4}x + \frac{6}{4}= \frac{1}{4}x + \frac{3}{2}$

The inverse is: $y= \frac{1}{4}x + \frac{3}{2}$

Find the inverse of the following algebraic function:

$f(x)=\frac{1}{x-3}$

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

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

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

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

Explanation

To find the inverse, switch the placement of the and variables:

$f(x)=\frac{1}{x-3}$

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

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

Next, should be isolated, providing the inverse function:

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

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

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

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