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3 years ago

Let f(x) = x2 + x + 2 and g(x) = 2x + 5. Find f(g(x)). Show each step of your work. (10 points)

Mathematics

1 answer:

DaniilM [7]3 years ago

6 0

We were given;

f(x) = x² + x + 2

and

g(x) = 2x + 5

f(g(x)) = f(2x+5)

= (2x+5)²+(2x+5)+2

Expanding brackets gives;

=(2x)²+2(2x)(5) +5²+(2x+5)+2

Simplifying gives;

= 4x² +20x+25+ 2x+5+2

f(g(x))= 4x² +22x +32

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3 years ago

Read 2 more answers

A population has a mean of 200 and a standard deviation of 50. Suppose a sample of size 100 is selected and x is used to estimat

zmey [24]

Answer:

a) 0.6426 = 64.26% probability that the sample mean will be within +/- 5 of the population mean.

b) 0.9544 = 95.44% probability that the sample mean will be within +/- 10 of the population mean.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this question, we have that:

$\mu = 200, \sigma = 50, n = 100, s = \frac{50}{\sqrt{100}} = 5$

a. What is the probability that the sample mean will be within +/- 5 of the population mean (to 4 decimals)?

This is the pvalue of Z when X = 200 + 5 = 205 subtracted by the pvalue of Z when X = 200 - 5 = 195.

Due to the Central Limit Theorem, Z is:

$Z = \frac{X - \mu}{s}$

X = 205

$Z = \frac{X - \mu}{s}$

$Z = \frac{205 - 200}{5}$

$Z = 1$

$Z = 1$ has a pvalue of 0.8413.

X = 195

$Z = \frac{X - \mu}{s}$

$Z = \frac{195 - 200}{5}$

$Z = -1$

$Z = -1$ has a pvalue of 0.1587.

0.8413 - 0.1587 = 0.6426

0.6426 = 64.26% probability that the sample mean will be within +/- 5 of the population mean.

b. What is the probability that the sample mean will be within +/- 10 of the population mean (to 4 decimals)?

This is the pvalue of Z when X = 210 subtracted by the pvalue of Z when X = 190.

X = 210

$Z = \frac{X - \mu}{s}$

$Z = \frac{210 - 200}{5}$

$Z = 2$

$Z = 2$ has a pvalue of 0.9772.

X = 195

$Z = \frac{X - \mu}{s}$

$Z = \frac{190 - 200}{5}$

$Z = -2$

$Z = -2$ has a pvalue of 0.0228.

0.9772 - 0.0228 = 0.9544

0.9544 = 95.44% probability that the sample mean will be within +/- 10 of the population mean.

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3 years ago

What single decimal multiplier would you use to increase by 18% followed by a 12% decrease?

ValentinkaMS [17]

Answer:

The single decimal multiplier used to increase by 18% followed by a 12% decrease is 0.21.

Step-by-step explanation:

Let 'x' be the number

As first the number has to be increased by 18%

Step 1: write the percentage '18%' in decimal form

18% = 0.18

Step 2: For percentage increases: add the decimal to 1

1 + 0.18 = 1.18

Step 3: Multiply the number x by the multiplier, found in Step 2

x × 1.18 = 1.18x

NEXT WE HAVE TO DECREASE 1.18x by 12%

Apply the same method but with a number 1.18x

Write the percentage '12%' in decimal form

12% = 0.12

For percentage decreases: subtract the decimal to 1

1 - 0.18 = 0.18

Multiply the number 1.18x by the multiplier, found in Previous step

1.18x × 0.18 = 0.21x

Therefore, the single decimal multiplier used to increase by 18% followed by a 12% decrease is 0.21.

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Answer:

1. y = $\frac{-A}{By}$ x + $\frac{C}{By}$

2. Divide the Opposite of A and B

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Step-by-step explanation:

1. Ax + By = C

By = -Ax + C

y = $\frac{-A}{By}$ x + $\frac{C}{By}$

2. No explanation

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