Question
Find left-parenthesis f plus g right-parenthesis left-parenthesis x right-parenthesis when lower f left-parenthesis x right-parenthesis equals x cubed minus 2 x squared plus 1 and lower g left-parenthesis x right-parenthesis equals 4 x cubed minus 5 x plus 7.
Answers
Answer
Find left-parenthesis f minus g right-parenthesis left-parenthesis x right-parenthesis when lower f left-parenthesis x right-parenthesis equals Start Fraction 2 x plus 6 over 3 x End Fraction and lower g left-parenthesis x right-parenthesis equals Start Fraction Start Root x End Root minus 8 over 3 x End Fraction.
Answer
To convert temperatures from degrees Celsius to degrees Fahrenheit, you can use the function upper F left-parenthesis x right-parenthesis equals Start Fraction 9 over 5 End Fraction x plus 32 where x is the temperature in degrees Celsius. To convert temperatures from degrees Kelvin to degrees Celsius, you can use the function upper C left-parenthesis x right-parenthesis equals x minus 273.15, where x is the temperature in degrees Kelvin. Find the composite function that can be used to convert temperatures from degrees Kelvin to degrees Fahrenheit.
Answer
Determine the domain of the function left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis where lower f left-parenthesis x right-parenthesis equals Start Fraction 3 x minus 1 over x minus 4 End Fraction and lower g left-parenthesis x right-parenthesis equals Start Fraction x plus 1 over x End Fraction.
Answer
choices
a) left-parenthesis negative infinity comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesisImage with alt text: left-parenthesis negative infinity comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
b) left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
c) left-parenthesis negative infinity comma 4 right-parenthesis union left-parenthesis 4 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 4 right-parenthesis union left-parenthesis 4 comma infinity right-parenthesis
d) left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
a) left-parenthesis negative infinity comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesisImage with alt text: left-parenthesis negative infinity comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
b) left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis 0 comma one-third right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
c) left-parenthesis negative infinity comma 4 right-parenthesis union left-parenthesis 4 comma infinity right-parenthesis
Image with alt text: left-parenthesis negative infinity comma 4 right-parenthesis union left-parenthesis 4 comma infinity right-parenthesis
d) left-parenthesis negative infinity comma 0 right-parenthesis union left-parenthesis one-third comma infinity right-parenthesis
Answer
Find left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis when lower f left-parenthesis x right-parenthesis equals Start Fraction 2 over x plus 3 End Fraction and lower g left-parenthesis x right-parenthesis equals Start Fraction 1 over 2x End Fraction.
Answer
choices
a) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 4 x over 1 plus 6 x End FractionImage with alt text: left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 4 x over 1 plus 6 x End Fraction
b) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 1 over x squared plus 3 x End Fraction
Image with alt text: left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 1 over x squared plus 3 x End Fraction
c) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction x plus 3 over 4 End Fraction
Image with alt text: left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction x plus 3 over 4 End Fraction
d) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 1 over x plus 3 End Fraction
a) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 4 x over 1 plus 6 x End FractionImage with alt text: left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 4 x over 1 plus 6 x End Fraction
b) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 1 over x squared plus 3 x End Fraction
Image with alt text: left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 1 over x squared plus 3 x End Fraction
c) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction x plus 3 over 4 End Fraction
Image with alt text: left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction x plus 3 over 4 End Fraction
d) left-parenthesis lower f circle lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Fraction 1 over x plus 3 End Fraction
Answered by
GPT 3.5
To find \( f(x) + g(x) \), we first need to substitute the given expressions for \( f(x) \) and \( g(x) \) into the expression \( f(x) + g(x) \).
Given:
\( f(x) = x^3 - 2x^2 + 1 \)
\( g(x) = 4x^3 - 5x + 7 \)
Substituting these values into the expression:
\( f(x) + g(x) = (x^3 - 2x^2 + 1) + (4x^3 - 5x + 7) \)
Next, combine like terms by adding the coefficients of each term:
\( f(x) + g(x) = (1 + 7) + (x^3 + 4x^3) + (-2x^2 - 5x) \)
Simplifying the coefficients:
\( f(x) + g(x) = 8 + 5x^3 - 2x^2 - 5x \)
Thus, \( f(x) + g(x) = 5x^3 - 2x^2 - 5x + 8 \).
Given:
\( f(x) = x^3 - 2x^2 + 1 \)
\( g(x) = 4x^3 - 5x + 7 \)
Substituting these values into the expression:
\( f(x) + g(x) = (x^3 - 2x^2 + 1) + (4x^3 - 5x + 7) \)
Next, combine like terms by adding the coefficients of each term:
\( f(x) + g(x) = (1 + 7) + (x^3 + 4x^3) + (-2x^2 - 5x) \)
Simplifying the coefficients:
\( f(x) + g(x) = 8 + 5x^3 - 2x^2 - 5x \)
Thus, \( f(x) + g(x) = 5x^3 - 2x^2 - 5x + 8 \).
Answered by
GPT 3.5
To find \( f(x) - g(x) \), we first need to substitute the given expressions for \( f(x) \) and \( g(x) \) into the expression \( f(x) - g(x) \).
Given:
\( f(x) = \frac{2x+6}{3x} \)
\( g(x) = \frac{\sqrt{x} - 8}{3x} \)
Substituting these values into the expression:
\( f(x) - g(x) = \frac{2x+6}{3x} - \frac{\sqrt{x} - 8}{3x} \)
To subtract these fractions, we need a common denominator, which is \( 3x \):
\( f(x) - g(x) = \frac{(2x+6) - (\sqrt{x} - 8)}{3x} \)
Simplifying the numerator:
\( f(x) - g(x) = \frac{2x+6 - \sqrt{x} + 8}{3x} \)
Combining like terms:
\( f(x) - g(x) = \frac{2x + 8 - \sqrt{x} + 6}{3x} \)
Simplifying further:
\( f(x) - g(x) = \frac{2x + 14 - \sqrt{x}}{3x} \)
Thus, \( f(x) - g(x) = \frac{2x + 14 - \sqrt{x}}{3x} \)
Given:
\( f(x) = \frac{2x+6}{3x} \)
\( g(x) = \frac{\sqrt{x} - 8}{3x} \)
Substituting these values into the expression:
\( f(x) - g(x) = \frac{2x+6}{3x} - \frac{\sqrt{x} - 8}{3x} \)
To subtract these fractions, we need a common denominator, which is \( 3x \):
\( f(x) - g(x) = \frac{(2x+6) - (\sqrt{x} - 8)}{3x} \)
Simplifying the numerator:
\( f(x) - g(x) = \frac{2x+6 - \sqrt{x} + 8}{3x} \)
Combining like terms:
\( f(x) - g(x) = \frac{2x + 8 - \sqrt{x} + 6}{3x} \)
Simplifying further:
\( f(x) - g(x) = \frac{2x + 14 - \sqrt{x}}{3x} \)
Thus, \( f(x) - g(x) = \frac{2x + 14 - \sqrt{x}}{3x} \)
Answered by
GPT 3.5
To find the composite function that converts temperatures from degrees Kelvin to degrees Fahrenheit, we need to substitute the function \( C(x) = x - 273.15 \) into the function \( F(x) = \frac{9}{5}x + 32 \).
Substituting \( C(x) \) into \( F(x) \), we get:
\( F(C(x)) = \frac{9}{5}(x - 273.15) + 32 \)
Simplifying:
\( F(C(x)) = \frac{9}{5}x - \frac{9}{5}(273.15) + 32 \)
Further simplification:
\( F(C(x)) = \frac{9}{5}x - 491.67 + 32 \)
Combining constants:
\( F(C(x)) = \frac{9}{5}x - 459.67 \)
Therefore, the composite function that converts temperatures from degrees Kelvin to degrees Fahrenheit is:
\( F(C(x)) = \frac{9}{5}x - 459.67 \)
Substituting \( C(x) \) into \( F(x) \), we get:
\( F(C(x)) = \frac{9}{5}(x - 273.15) + 32 \)
Simplifying:
\( F(C(x)) = \frac{9}{5}x - \frac{9}{5}(273.15) + 32 \)
Further simplification:
\( F(C(x)) = \frac{9}{5}x - 491.67 + 32 \)
Combining constants:
\( F(C(x)) = \frac{9}{5}x - 459.67 \)
Therefore, the composite function that converts temperatures from degrees Kelvin to degrees Fahrenheit is:
\( F(C(x)) = \frac{9}{5}x - 459.67 \)
Answered by
GPT 3.5
To determine the domain of the function \( f \circ g \), we need to consider the domain of \( g(x) \) and ensure that the output of \( g(x) \) falls within the domain of \( f(x) \).
Given:
\( f(x) = \frac{3x - 1}{x - 4} \)
\( g(x) = \frac{x + 1}{x} \)
To determine the domain of \( g(x) \), we need to find the values of \( x \) for which the denominator is not equal to zero.
In this case, we have \( x \neq 0 \), since division by zero is undefined.
Now, to find the domain of \( f \circ g \), we need to ensure that the output of \( g(x) \) falls within the domain of \( f(x) \), i.e., the denominator \( x - 4 \) should not be zero.
If \( x - 4 = 0 \), then \( x = 4 \).
Therefore, the domain of the function \( f \circ g \) is all real numbers \( x \) except \( x = 0 \) and \( x = 4 \).
Given:
\( f(x) = \frac{3x - 1}{x - 4} \)
\( g(x) = \frac{x + 1}{x} \)
To determine the domain of \( g(x) \), we need to find the values of \( x \) for which the denominator is not equal to zero.
In this case, we have \( x \neq 0 \), since division by zero is undefined.
Now, to find the domain of \( f \circ g \), we need to ensure that the output of \( g(x) \) falls within the domain of \( f(x) \), i.e., the denominator \( x - 4 \) should not be zero.
If \( x - 4 = 0 \), then \( x = 4 \).
Therefore, the domain of the function \( f \circ g \) is all real numbers \( x \) except \( x = 0 \) and \( x = 4 \).
Answered by
GPT 3.5
The correct answer is:
b) \( \left(-\infty, 0\right) \cup \left(0, \frac{1}{3}\right) \cup \left(\frac{1}{3}, \infty\right) \)
b) \( \left(-\infty, 0\right) \cup \left(0, \frac{1}{3}\right) \cup \left(\frac{1}{3}, \infty\right) \)
Answered by
GPT 3.5
To find \( f \circ g \), we need to substitute the function \( g(x) \) into the function \( f(x) \).
Given:
\( f(x) = \frac{2}{x + 3} \)
\( g(x) = \frac{1}{2x} \)
To find \( f \circ g \), we substitute \( g(x) \) into \( f(x) \):
\( f(g(x)) = f\left(\frac{1}{2x}\right) \)
Substituting \( g(x) \) into \( f(x) \):
\( f(g(x)) = \frac{2}{\frac{1}{2x} + 3} \)
Simplifying:
\( f(g(x)) = \frac{2}{\frac{1 + 6x}{2x}} \)
Inverting the denominator:
\( f(g(x)) = \frac{2x}{1 + 6x} \)
Therefore, \( f \circ g(x) = \frac{2x}{1 + 6x} \)
Given:
\( f(x) = \frac{2}{x + 3} \)
\( g(x) = \frac{1}{2x} \)
To find \( f \circ g \), we substitute \( g(x) \) into \( f(x) \):
\( f(g(x)) = f\left(\frac{1}{2x}\right) \)
Substituting \( g(x) \) into \( f(x) \):
\( f(g(x)) = \frac{2}{\frac{1}{2x} + 3} \)
Simplifying:
\( f(g(x)) = \frac{2}{\frac{1 + 6x}{2x}} \)
Inverting the denominator:
\( f(g(x)) = \frac{2x}{1 + 6x} \)
Therefore, \( f \circ g(x) = \frac{2x}{1 + 6x} \)
Answered by
GPT 3.5
The correct answer is:
a) \( (f \circ g)(x) = \frac{4x}{1 + 6x} \)
a) \( (f \circ g)(x) = \frac{4x}{1 + 6x} \)
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