Which of the following functions has an inverse that is NOT a function?

A) f(x) = (1/2)x - 1/2
B) f(x) = (x - 1)^3 + 2
C) f(x) = 2^x
D) f(x) = x(x - 1)

Step-by-step explanation:

Content

Functions and their inverses

We begin with a simple example.

Example

Let f(x)=2x and g(x)=x2.

Apply the function g to the number 3, and then apply f to the result:

g(3)=32andf(32)=3.

A similar thing happens if we first apply f and then apply g:

f(3)=6andg(6)=3.

It is clear that this will happen with any starting number. This is expressed as

f(g(x))g(f(x))=x,for all x=x,for all x.

The function f reverses the effect of g, and the function g reverses the effect of f. We say that f and g are inverses of each other.

As another example, we have

(x−−√3)3=xandx3−−√3=x,

for all real x. So the functions f(x)=x3 and g(x)=x−−√3 are inverses of each other.

If x≥0, then (x−−√)2=x and x2−−√=x. If x<0, then x−−√ is not defined. So the functions f(x)=x2 and g(x)=x−−√ are inverses of each other, but we need to be careful about domains. We will look at this more carefully later in this section.

Basics

In an earlier section of this module, we defined the composite of two functions h and g by (g∘h)(x)=g(h(x)).

Definitions

The zero function 0–:R→R is defined by 0–(x)=0, for all x.

The identity function id:R→R is defined by id(x)=x, for all x.

Example

Consider a function f:R→R.

Prove that

0–∘f=0–

f∘id=f

id∘f=f.

Show that f∘0– does not necessarily equal 0–.

Solution

We have (0–∘f)(x)=0–(f(x))=0, for all x, and so 0–∘f=0–.

We have (f∘id)(x)=f(id(x))=f(x), for all x, and so f∘id=f.

We have (id∘f)(x)=id(f(x))=f(x), for all x, and so id∘f=f.

Consider the function given by f(x)=2, for all x. Then f∘0–(x)=f(0–(x))=f(0)=2, and so f∘0–≠0–.

Definition

Let f be a function with both domain and range all real numbers. Then the function g is the inverse of f if

f(g(x))g(f(x))=x,for all x,and=x,for all x.

That is, f∘g=id and g∘f=id.

Notes.

Clearly, if g is the inverse of f, then f is the inverse of g.

We denote the inverse of f by f−1. We read f−1 as 'f inverse'. Note that f inverse has nothing to do with the function 1f.

Example

Let f(x)=x+2 and let g(x)=x−2. Show that f and g are inverses of each other.

Solution

We have

f(g(x))=f(x−2)=x−2+2=x,for all x(f∘g=id)

and

g(f(x))=g(x+2)=x+2−2=x,for all x(g∘f=id).

Hence, the functions f and g are inverses of each other.

Exercise 5

Find the inverse of

f(x)=x+7

f(x)=4x+5.

Example

Let f(x)=ax+b with a≠0. Find the inverse of f.

Solution

We have x=f(x)−ba, for all x. So let g(x)=x−ba. Then

f(g(x))g(f(x))=f(x−ba)=a(x−ba)+b=x=g(ax+b)=(ax+b)−ba=x,

for all x. Hence, g is the inverse of f.

Exercise 6

Show that f(x)=x5 and g(x)=x15 are inverses of each other.

Find the inverse of f(x)=x3+2.

We do not yet have a general enough concept of inverses, since x2 and x−−√ do not fit into this framework, nor do ex and logex. We will give a definition that covers these functions later in this section.

The horizontal-line test

Consider the function f(x)=x2, which has domain the reals and range A={x:x≥0}. Does f have an inverse?

The following graph shows that it does not. We have f(−2)=f(2)=4, and so f−1(4) would have to take two values, −2 and 2! Hence, f does not have an inverse.

Graph of y = x squared and the line y = 4 on the one set of axes.

This idea can be formulated as a test.

Horizontal-line test

Let f be a function. If there is a horizontal line y=c that meets the graph y=f(x) at more than one point, then f does not have an inverse.

Notes. Remember that the vertical-line test determines whether a relation is a function.

Example

Consider the function

f(x)=x3−x=(x+1)x(x−1).

Its graph is shown in the following diagram.

Graph of y = x cubed minus x.

Does f have an inverse?

Solution

The line y=0 meets the graph at three points. By the horizontal-line test, the function f does not have an inverse.

The function whose inverse does not exist is f(x) = x(x - 1)

The correct option is (D)  f(x) = x(x - 1)

An inverse is a function that serves to “undo” another function. That is, if f(x) produces y, then putting y into the inverse of f produces the output x.

First,  f(x)= [tex]\frac{1}{2} x -\frac{1}{2}[/tex]

let y= [tex]\frac{1}{2} x -\frac{1}{2}[/tex]

On solving for x we get a unique value

Then replace x and y.

It shows that the function have a unique value, which satisfies the condition of inverse.

Now, f(x) =[tex](x - 1)^3 + 2[/tex]

Again, solving for y we can get a cube root function which is a inverse of cube.

Hence, the inverse of [tex](x - 1)^3 + 2[/tex] exists.

Next, f(x) =[tex]2^x[/tex]

Solving for above we get logarithmic value. Log function are inverse of exponential function.

Hence, the inverse of [tex]2^x[/tex] exists.

Last, f(x)= x(x-1)

Solving for above create a square value.

The inverse of square never exist because having square root gives two value one is positive and other is negative.

Hence, the inverse of x(x-1) not exists.

Hence the function whose inverse does not exist is x(x-1).

Learn more about inverse of function here:

https://brainly.com/question/2541698

#SPJ2

50×54+35-44
Simplification
no spam
steps must​

Noah bought 15 baseball cards for $9.00 assuming each baseball card cost the same amount answer the following questions

A salesperson earns an income of $600 per week aus an dditional 5% commission on her total sales for that weck a. Write an equation that models the Salespersons total weekly income. b. What would be the salespersons income if her total sales were $10.000 that week? c. How much would a salesperson have to sell to have a $2000 weekly​

Trong 300 người bệnh chỉ dùng thuốc A có 55% người khỏi; trong 240 người bệnh chỉ dùng thuốc B có 50% người khỏi. Với mức ý nghĩa 5% có thể nói thuốc A hiệu quả hơn thuốc B không?

b. What is the number that is halfway between the two values?
7 and 11
Submit

Dell Computers receives large shipments of microprocessors (chips) from Intel Corp. It must try to ensure that the proportion of defective chips is small and has contracted to keep the proportion of defective chips at 5% or less. Suppose that Dell tests 25 chips (with replacement) out of a shipment of thousands.
a) What is the probability of obtaining at least three defects in the sample, if the entire shipment has 5% defectives? If a sample results in at least three defects, does it appear that Intel has provided a shipment which is consistent with the contract? Briefly explain, using the calculated probability.
b) For the shipment described in part a, find the expected number of sampled chips that are defective and the standard deviation of the number of sampled chips that are defective.
c) Suppose that 20% of the entire shipment is defective. What is the probability that more than 17 chips in the sample are good (not defective)?​

solve pls Brainliest

[tex]\frac { a } { ( a - b ) ( a - c ) } + \frac { b } { ( b - c ) ( b - a ) } + \frac { c } { ( c - a ) ( c - b ) }[/tex]
the formula is the question pleaseee helppp quickkk
​

Heyyyyyyyyyyyy can you help

covert 31 days into minutes

Three workers worked for 5 hours, 7 hours and 12 hours. The total wage of $
13200 was divided among the three workers according to the number of hours of
work they did. How much did each get?

Write the equation of the line, given the y and x-intercepts. a x-intercept = –68, y-intercept = –34

Find the Value of x

Please answer!!
No links!!

Suppose there is a 1.2 drop in temperature for every thousand feet that an airplane climbs into the sky. If the temperature on the ground is ​57.8, what will be the temperature when the plane reaches an altitude of ​7000?

Using the graph of y = f(x) to the right, and given that the zero of the function is x=20, find the following.

Help!!!! Jack walked 3 1/3 miles in 4/3 hours. If he walked at a constant rate, how far did he walk in 15 minutes?

QUICK!!

fill in the blanks below in order to justify whether or not the mapping shown represents a function.

solve pls brainliest

Given g(x) = 2x + 5, find g(1).

Lillie is saving up for a trip she is taking with friends during her break from school. If Lillie's current monthly net pay is $429.00 and her monthly expenses are $347.49, what percent of her net pay is left for savings?

a
19%

b
23%

c
65%

d
81%

On Friday, 0.325 of the students in
Keisha's math class were absent. What
percent of the students were absent?
Look at picture for choices!! Quick!!

Andre wrote the expression -2 + 4x = 3 to represent the relationship shown in the table. Write two other expressions that also represent the relationship shown in the table. ​

The lengths of the worms (in inches) are listed here.
Which line plot shows this data?

2. Louis kept losing money through a hole in his pocket. He started with 35€, lost 20€.
put in 754 more, spent 43¢ lost 16¢ again, and then put in 14¢. How much change should
there be in his pocket?

5(Y-1)=3(2Y-5)-(1-3Y)

Wooden flooring costs £35 per square metre. The floor is then edged with a narrow wooden strip which costs £4 per metre.

How much does it cost for a 6m by 4m room, including the edging?

direct variation
Hooke's Law for an elastic spring states that the distance a spring stretches varies directly as the force applied. if a force of 180 newtons stretches a spring 5 cm, how much will a force of 270 newtons stretch the same spring?

pls help (;ŏ﹏ŏ)​

Find the slope between these two points:
(4,7).(-6, -4)

150 miles on 4 gallons of gas. How many miles for 1 gallon of gas?

You deposit $400 in an account earning 8% interest compounded annually. How much will you have in the account in 15 years?