Question Determine which pairs of functions are equivalent. Explain your reasoning. a) $f(x)=2x^2+x-7$ and $g(x)=\frac{6x^2+3x-21}{3}$ b) $h(x)=3x^2+5x+1$ and $j(x)=\frac{3x^3+5x^2+x}{x}, x \neq 0$

Studdy Solution

STEP 1

Assumptions
1. We are given two pairs of functions.
2. We need to determine if the functions in each pair are equivalent.
3. Equivalent functions will have the same value for all corresponding inputs in their domains.

STEP 2

We will start with the first pair of functions, $f(x)$ and $g(x)$ .
$f(x) = 2x^2 + x - 7$ $g(x) = \frac{6x^2 + 3x - 21}{3}$

STEP 3

Simplify the function $g(x)$ by distributing the division across the terms in the numerator.
$g(x) = \frac{6x^2}{3} + \frac{3x}{3} - \frac{21}{3}$

STEP 4

Perform the division for each term in $g(x)$ .
$g(x) = 2x^2 + x - 7$

STEP 5

Compare the simplified form of $g(x)$ with $f(x)$ .
$f(x) = 2x^2 + x - 7$ $g(x) = 2x^2 + x - 7$

STEP 6

Since $f(x)$ and $g(x)$ are identical after simplification, we conclude that $f(x)$ and $g(x)$ are equivalent.

STEP 7

Now, let's move on to the second pair of functions, $h(x)$ and $j(x)$ .
$h(x) = 3x^2 + 5x + 1$ $j(x) = \frac{3x^3 + 5x^2 + x}{x}, x \neq 0$

STEP 8

Simplify the function $j(x)$ by canceling the common factor of $x$ in the numerator and the denominator, noting that $x \neq 0$ .
$j(x) = \frac{x(3x^2 + 5x + 1)}{x}$

STEP 9

Perform the cancellation in $j(x)$ .
$j(x) = 3x^2 + 5x + 1$

STEP 10

Compare the simplified form of $j(x)$ with $h(x)$ .
$h(x) = 3x^2 + 5x + 1$ $j(x) = 3x^2 + 5x + 1$

STEP 11

Since $h(x)$ and $j(x)$ are identical after simplification, we conclude that $h(x)$ and $j(x)$ are equivalent.

STEP 12

State the conclusion for both pairs of functions.
The functions $f(x)$ and $g(x)$ are equivalent, as are the functions $h(x)$ and $j(x)$ . This is because, after simplification, the expressions for each pair are identical.

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Question Determine which pairs of functions are equivalent. Explain your reasoning. a) $f(x)=2x^2+x-7$ and $g(x)=\frac{6x^2+3x-21}{3}$ b) $h(x)=3x^2+5x+1$ and $j(x)=\frac{3x^3+5x^2+x}{x}, x \neq 0$

Studdy Solution

STEP 1

Assumptions
1. We are given two pairs of functions.
2. We need to determine if the functions in each pair are equivalent.
3. Equivalent functions will have the same value for all corresponding inputs in their domains.

STEP 2

We will start with the first pair of functions, $f(x)$ and $g(x)$ .
$f(x) = 2x^2 + x - 7$ $g(x) = \frac{6x^2 + 3x - 21}{3}$

STEP 3

Simplify the function $g(x)$ by distributing the division across the terms in the numerator.
$g(x) = \frac{6x^2}{3} + \frac{3x}{3} - \frac{21}{3}$

STEP 4

Perform the division for each term in $g(x)$ .
$g(x) = 2x^2 + x - 7$

STEP 5

Compare the simplified form of $g(x)$ with $f(x)$ .
$f(x) = 2x^2 + x - 7$ $g(x) = 2x^2 + x - 7$

STEP 6

Since $f(x)$ and $g(x)$ are identical after simplification, we conclude that $f(x)$ and $g(x)$ are equivalent.

STEP 7

Now, let's move on to the second pair of functions, $h(x)$ and $j(x)$ .
$h(x) = 3x^2 + 5x + 1$ $j(x) = \frac{3x^3 + 5x^2 + x}{x}, x \neq 0$

STEP 8

Simplify the function $j(x)$ by canceling the common factor of $x$ in the numerator and the denominator, noting that $x \neq 0$ .
$j(x) = \frac{x(3x^2 + 5x + 1)}{x}$

STEP 9

Perform the cancellation in $j(x)$ .
$j(x) = 3x^2 + 5x + 1$

STEP 10

Compare the simplified form of $j(x)$ with $h(x)$ .
$h(x) = 3x^2 + 5x + 1$ $j(x) = 3x^2 + 5x + 1$

STEP 11

Since $h(x)$ and $j(x)$ are identical after simplification, we conclude that $h(x)$ and $j(x)$ are equivalent.

STEP 12

State the conclusion for both pairs of functions.
The functions $f(x)$ and $g(x)$ are equivalent, as are the functions $h(x)$ and $j(x)$ . This is because, after simplification, the expressions for each pair are identical.

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Studdy Solution

STEP 1

Assumptions1. We are given two pairs of functions.2. We need to determine if the functions in each pair are equivalent.3. Equivalent functions will have the same value for all corresponding inputs in their domains.

STEP 2

We will start with the first pair of functions, f(x)f(x)f(x) and g(x)g(x)g(x).f(x)=2x2+x−7f(x) = 2x^2 + x - 7f(x)=2x2+x−7 g(x)=6x2+3x−213g(x) = \frac{6x^2 + 3x - 21}{3}g(x)=36x2+3x−21​

STEP 3

Simplify the function g(x)g(x)g(x) by distributing the division across the terms in the numerator.g(x)=6x23+3x3−213g(x) = \frac{6x^2}{3} + \frac{3x}{3} - \frac{21}{3}g(x)=36x2​+33x​−321​

STEP 4

Perform the division for each term in g(x)g(x)g(x).g(x)=2x2+x−7g(x) = 2x^2 + x - 7g(x)=2x2+x−7

STEP 5

Compare the simplified form of g(x)g(x)g(x) with f(x)f(x)f(x).f(x)=2x2+x−7f(x) = 2x^2 + x - 7f(x)=2x2+x−7 g(x)=2x2+x−7g(x) = 2x^2 + x - 7g(x)=2x2+x−7

STEP 6

Since f(x)f(x)f(x) and g(x)g(x)g(x) are identical after simplification, we conclude that f(x)f(x)f(x) and g(x)g(x)g(x) are equivalent.

STEP 7

Now, let's move on to the second pair of functions, h(x)h(x)h(x) and j(x)j(x)j(x).h(x)=3x2+5x+1h(x) = 3x^2 + 5x + 1h(x)=3x2+5x+1 j(x)=3x3+5x2+xx,x≠0j(x) = \frac{3x^3 + 5x^2 + x}{x}, x \neq 0j(x)=x3x3+5x2+x​,x=0

STEP 8

Simplify the function j(x)j(x)j(x) by canceling the common factor of xxx in the numerator and the denominator, noting that x≠0x \neq 0x=0.j(x)=x(3x2+5x+1)xj(x) = \frac{x(3x^2 + 5x + 1)}{x}j(x)=xx(3x2+5x+1)​

STEP 9

Perform the cancellation in j(x)j(x)j(x).j(x)=3x2+5x+1j(x) = 3x^2 + 5x + 1j(x)=3x2+5x+1

STEP 10

Compare the simplified form of j(x)j(x)j(x) with h(x)h(x)h(x).h(x)=3x2+5x+1h(x) = 3x^2 + 5x + 1h(x)=3x2+5x+1 j(x)=3x2+5x+1j(x) = 3x^2 + 5x + 1j(x)=3x2+5x+1

STEP 11

Since h(x)h(x)h(x) and j(x)j(x)j(x) are identical after simplification, we conclude that h(x)h(x)h(x) and j(x)j(x)j(x) are equivalent.

STEP 12

State the conclusion for both pairs of functions.The functions f(x)f(x)f(x) and g(x)g(x)g(x) are equivalent, as are the functions h(x)h(x)h(x) and j(x)j(x)j(x). This is because, after simplification, the expressions for each pair are identical.

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Assumptions
1. We are given two pairs of functions.
2. We need to determine if the functions in each pair are equivalent.
3. Equivalent functions will have the same value for all corresponding inputs in their domains.

We will start with the first pair of functions, $f(x)$ and $g(x)$ .
$f(x) = 2x^2 + x - 7$ $g(x) = \frac{6x^2 + 3x - 21}{3}$

Simplify the function $g(x)$ by distributing the division across the terms in the numerator.
$g(x) = \frac{6x^2}{3} + \frac{3x}{3} - \frac{21}{3}$

Perform the division for each term in $g(x)$ .
$g(x) = 2x^2 + x - 7$

Compare the simplified form of $g(x)$ with $f(x)$ .
$f(x) = 2x^2 + x - 7$ $g(x) = 2x^2 + x - 7$

Since $f(x)$ and $g(x)$ are identical after simplification, we conclude that $f(x)$ and $g(x)$ are equivalent.

Now, let's move on to the second pair of functions, $h(x)$ and $j(x)$ .
$h(x) = 3x^2 + 5x + 1$ $j(x) = \frac{3x^3 + 5x^2 + x}{x}, x \neq 0$

Simplify the function $j(x)$ by canceling the common factor of $x$ in the numerator and the denominator, noting that $x \neq 0$ .
$j(x) = \frac{x(3x^2 + 5x + 1)}{x}$

Perform the cancellation in $j(x)$ .
$j(x) = 3x^2 + 5x + 1$

Compare the simplified form of $j(x)$ with $h(x)$ .
$h(x) = 3x^2 + 5x + 1$ $j(x) = 3x^2 + 5x + 1$

Since $h(x)$ and $j(x)$ are identical after simplification, we conclude that $h(x)$ and $j(x)$ are equivalent.

State the conclusion for both pairs of functions.
The functions $f(x)$ and $g(x)$ are equivalent, as are the functions $h(x)$ and $j(x)$ . This is because, after simplification, the expressions for each pair are identical.