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Chapter 0: Problem 9

Use the two given functions to write y as a function of x. $$ y=w^{2}-2, w=x+3 $$

Short Answer

Expert verified
y = x^2 + 6x + 7

Step by step solution

01

- Substitute w in terms of x

Given the function for w: \[w = x + 3\] substitute this into the equation for y: \[y = w^2 - 2\]
02

- Apply the substitution

Substitute \(w = x + 3\) into \(y = w^2 - 2\): \[y = (x + 3)^2 - 2\]
03

- Expand the squared term

Expand the expression \[(x + 3)^2\]: \[(x + 3)^2 = x^2 + 6x + 9\]
04

- Simplify the expression

Substitute the expanded term back into the equation for y: \[y = x^2 + 6x + 9 - 2\] Combine like terms: \[y = x^2 + 6x + 7\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

function substitution
Function substitution is a method used in algebra to simplify complex equations. It involves replacing one variable with an expression that equals that variable.
In this exercise, we are given two functions: \( w = x + 3 \) and \( y = w^2 - 2 \).
To write \( y \) as a function of \( x \), we first need to substitute the expression for \( w \) into the equation for \( y \).

This step transforms \( y = w^2 - 2 \) into \( y = (x + 3)^2 - 2 \) by replacing \( w \) with \( x + 3 \).
This type of substitution helps to align equations with the same variable, making them easier to handle.
polynomial expansion
Polynomial expansion is the process of expanding expressions that are raised to a power.
In this case, we need to expand \((x + 3)^2\) to simplify the equation.

To do this, we use the formula for expanding a binomial square: \((a + b)^2 = a^2 + 2ab + b^2 \).

Applying this to our equation where \(a = x\) and \(b = 3\), we get:
\[(x + 3)^2 = x^2 + 2(x)(3) + 3^2\] which simplifies to \[x^2 + 6x + 9\].

This expanded form lets us see all the terms separately, which will be useful in the next step when we simplify the expression.
simplifying expressions
Simplifying expressions involves combining like terms to make an equation easier to read and work with.

After expanding \((x + 3)^2\) to get \(x^2 + 6x + 9\), we substitute this back into the equation for \( y \): \[y = x^2 + 6x + 9 - 2\].

Next, we combine like terms. In this case, the terms \(+9\) and \(-2\) are constants that can be combined:
\[y = x^2 + 6x + 7\].

Now, \( y \) is clearly written as a function of \( x \): \( y = x^2 + 6x + 7 \).

This final simplified form makes it easy to see the relationship between \( x \) and \( y \).

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Most popular questions from this chapter

Solve each problem. The number of unmarried-couple households, \(h\) (in millions), can be approximated using the equation $$ h=0.229 n+5.203 $$ where \(n\) is the number of years since 2000 (Census Bureau, www.census.gov). a. Find and interpret the \(n\) -intercept for the line. b. Find and interpret the \(h\) -intercept for the line.

Find the equation of a circle with center (0,1) and radius \(3 .\)

For each pair of variables determine whether \(a\) is a function of \(b\), \(b\) is a function of \(a\), or neither. \(a\) is any real number and \(b\) is the absolute value of that number.

Solve each problem. A right triangle has a hypotenuse of \(10 \mathrm{ft}\). If one leg is \(4 \mathrm{ft}\), then what is the length of the other leg?

Let \(f(x)=3 x^{2}-x, g(x)=4 x-2,\) and \(k(x)=|x+3|\) Find the following. $$ x, \text { if } g(x)=3 $$
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