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3 years ago

Algebraically find the inverse of a function

Mathematics

1 answer:

Savatey [412]3 years ago

6 0

9514 1404 393

Answer:

1) f⁻¹(x) = 6 ± 2√(x -1)

3) y = (x +4)² -2

5) y = (x -4)³ -4

Step-by-step explanation:

In general, swap x and y, then solve for y. Quadratics, as in the first problem, do not have an inverse function: the inverse relation is double-valued, unless the domain is restricted. Here, we're just going to consider these to be "solve for ..." problems, without too much concern for domain or range.

1) x = f(y)

x = (1/4)(y -6)² +1

4(x -1) = (y-6)² . . . . . . subtract 1, multiply by 4

±2√(x -1) = y -6 . . . . square root

y = 6 ± 2√(x -1) . . . . inverse relation

f⁻¹(x) = 6 ± 2√(x -1) . . . . in functional form

3) x = √(y +2) -4

x +4 = √(y +2) . . . . add 4

(x +4)² = y +2 . . . . square both sides

y = (x +4)² -2 . . . . . subtract 2

5) x = ∛(y +4) +4

x -4 = ∛(y +4) . . . . . subtract 4

(x -4)³ = y +4 . . . . . cube both sides

y = (x -4)³ -4 . . . . . . subtract 4

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Vlad1618 [11]

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3 years ago

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A survey is being conducted in a county where 62% of the voters are Democrats and 38% are Republican. (a) What is the probabilit

Inessa05 [86]

Answer:

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats

Step-by-step explanation:

For each voter, there are only two possible outcomes. Either the voter is a Democrat, or he is not. The probability of the voter being a Democrat is independent of other voters. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

62% of the voters are Democrats

This means that $p = 0.62$

(a) What is the probability that two independently surveyed voters would both be Democrats?

This is P(X = 2) when n = 2. So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 2) = C_{2,2}.(0.62)^{2}.(0.38)^{0} = 0.3844$

0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats

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3 years ago

A restaurant has two different seating options: a table and a family booth. A table can seat 4 people, and a family booth can se

krok68 [10]

Answer: 242 = 190 + 4t

Step-by-step explanation:

You know that the maximum capacity of the restaurant is 242 people, meaning that at most there can only be that many customers seated at that time. Normally, the equation would be 242 = 10b + 4t, but since you already know the number of booths, your work is cut in half, giving you 242 = 10(19) + 4t. The equation would be this because you have the capacity being equal to the number of tables x the number of people at each table and the number of booths x the number of people seated at each of them.

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Answer: I need a picture off the pool and measurements.

Step-by-step explanation: I need this in order to figure out the problem and give you a helpful answer.

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3 years ago

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A survey collected data on annual credit card charges in seven different categories of expenditures: transportation, groceries,

Andru [333]

Answer:

A. Null and alternative hypothesis:

$H_0: \mu_d=0\\\\H_a:\mu_d\neq 0$

B. Yes. At a significance level of 0.05, there is enough evidence to support the claim that there is signficant difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.

P-value = 0.00002

C. As the difference is calculated as (population 1 − population 2), being population 1: groceries and population 2: dinning out, and knowing there is evidence that the true mean difference is positive, we can say that the groceries annual credit card charge is higher than dinning out annual credit card charge.

The point estimate is the sample mean difference d=$840.

The 95% confidence interval for the mean difference between the population means is (490, 1190).

Step-by-step explanation:

This is a hypothesis test for the population mean.

The claim is that there is signficant difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.

Then, the null and alternative hypothesis are:

$H_0: \mu_d=0\\\\H_a:\mu_d\neq 0$

The significance level is 0.05.

The sample has a size n=42.

The sample mean is M=840.

As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=1123.

The estimated standard error of the mean is computed using the formula:

$s_M=\dfrac{s}{\sqrt{n}}=\dfrac{1123}{\sqrt{42}}=173.2827$

Then, we can calculate the t-statistic as:

$t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{840-0}{173.2827}=\dfrac{840}{173.2827}=4.848$

The degrees of freedom for this sample size are:

$df=n-1=42-1=41$

This test is a two-tailed test, with 41 degrees of freedom and t=4.848, so the P-value for this test is calculated as (using a t-table):

$\text{P-value}=2\cdot P(t>4.848)=0.00002$

As the P-value (0.00002) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

At a significance level of 0.05, there is enough evidence to support the claim that there is signficant difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.

We have to calculate a 95% confidence interval for the mean difference.

The t-value for a 95% confidence interval and 41 degrees of freedom is t=2.02.

The margin of error (MOE) can be calculated as:

$MOE=t\cdot s_M=2.02 \cdot 173.283=350$

Then, the lower and upper bounds of the confidence interval are:

$LL=M-t \cdot s_M = 840-350=490\\\\UL=M+t \cdot s_M = 840+350=1190$

The 95% confidence interval for the mean difference is (490, 1190).

7 0

3 years ago

Algebraically find the inverse of a function​

Algebraically find the inverse of a function