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.
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, \( f(x) + g(x) = 5x^3 - 2x^2 - 5x + 8 \).
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.
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, \( f(x) - g(x) = \frac{2x + 14 - \sqrt{x}}{3x} \)
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.
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:
\( F(C(x)) = \frac{9}{5}x - 459.67 \)
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.
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 \( x = 4 \).
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
The correct answer is:
b) \( \left(-\infty, 0\right) \cup \left(0, \frac{1}{3}\right) \cup \left(\frac{1}{3}, \infty\right) \)