Convert this rational number

Aleks04 [339]

The answer is: $f^{-1} = 4x^2-3,\quad\text{for } x \leq 0$

The inverse of a function $f(x)$ is another function, $f^{-1}(x)$ , with the following property:

$f(f^{-1}(x)) = f^{-1}(f(x)) = x$

In other words, the inverse of a function does exactly "the opposite" of what the original function does, and so if you compute them both in sequence you return to the starting point.

Think for example to a function that doubles the input, $f(x)=2x$ , and one that halves it: $f(x)= \frac{x}{2}$ . Their composition is clearly the identity function $f(x)=x$ , since you consider "twice the half of something", or "half the double of something".

In general, to invert a function $y=f(x)$ , you have to solve the expression for $x$ , writing an expression like $x = g(y)$ . If you manage to do so, then $g$ is the inverse of $f$ .

In your case, you have

$f(x) = y = -\frac{1}{2}\sqrt{x+3}$

Multiply both sides by $-2$ to get

$-2y = \sqrt{x+3}$

Square both sides to get

$4y^2 = x+3$

Finally, subtract 3 from both sides to get

$x = 4y^2 - 3$

Since the name of the variables doesn't really have a meaning, you can say that the inverse function is

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

As for the domain of the inverse function, remember what we said ad the beginning: if the original function goes from set A (domain) to set B (codomain), then the inverse function goes from set B (domain) to set A (codomain). This means that the inverse function is defined on an element in B if and only if that element belongs to the range of the original function, i.e. the set of the elements of the codomain $b \in B$ such that there exists $a \in A : f(a)=b$ . So, we need the range of $f(x)$ .

We know that the range of $g(x)=\sqrt{x}$ is $[0,\infty)$ . When you transform it to $g(x)=\sqrt{x+3}$ you simply translate the graph horizontally, so the range doesn't change. But when you multiply the function times $-\frac{1}{2}$ you affect both extrema of the range, turning it into $(-\infty,0]$ , which you can simply write as $x \leq 0$

4 0

3 years ago

Suppose you can work at most a total of 25 hours per week. Baby-sitting, x, pays $6 per hour and working at the grocery store, y

Andrei [34K]

Answer:

y=7 number of hours at grocery store

x=18 number of hours at baby- sitting

Step-by-step explanation:

According to the information provided.

x is number of hours at baby- sitting

y is number of hours at grocery store

total number of hours worked

1) x+y =25

total earn in a week

2) x*$6 + y* $9 = $171

from equation 1

x+y=25

x= 25-y

we place the above derived equation in equation 2

x*$6 + y* $9 = $171

(25-y)*$6 + y* $9 = $171

(25*6) -6y +9y =171

150+3y=171

3y=171-150

3y=21

y=7 number of hours at grocery store

x= 25-y

x= 25-7

x=18 number of hours at baby- sitting

3 0

3 years ago

Please answer this question. FOLLOW ALL OF THE INSTRUCTIONS to get the right anser please. THANK YOU!

Alexus [3.1K]

A) It says in fraction form. y=5/4x +15
b) it says it should be a decimal. If you extend the line, it will reach 15.5 hours.
c) I think your answer was correct

7 0

3 years ago

The cost of going to the big playland park for a family is given by C=15x + 50, where C represents the cost and x represents the

IgorC [24]

Set the equation to 125.

15x+50= 125

Subtract 50.

15x= 75

Divide by 15 into both sides.

x= 5

5 family members went.

I hope this helps!
~cupcake

3 0

3 years ago

Identify the expression equivalent to 4(x + x + 7) − 2x + 8 − 4 by substituting x = 1 and x = 2. 6x + 11 3(x + 7) 2(3x + 16) 3x

sveta [45]

2(3x+16)

Step-by-step explanation:

Identify the expression equivalent to 4(x + x + 7) − 2x + 8 − 4

4(x + x + 7) − 2x + 8 − 4

When x=1 , the expression becomes 4(1+1+7)-2(1)+8-4=38

When x=2 , the expression becomes 4(2+2+7)-2(1)+8-4=44

Plug in x=1 and check with each expression

6x + 11 =6(1) +11= 16

3(x + 7) = 3(1+7)= 3(8)= 24

2(3x + 16) = 2(3(1)+16)= 38 , we got same answer when x=1, now check with x=2

2(3x + 16) = 2(3(2)+16)= 44

3x + 16= 3+16= 19

5 0

3 years ago