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Sagot :
To find the inverse of the function [tex]\( f(x) = \frac{19}{x^3} \)[/tex], follow these steps:
1. Start with the function [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = \frac{19}{x^3} \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse. Set [tex]\( x \)[/tex] equal to the function and replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[ x = \frac{19}{y^3} \][/tex]
3. Solve for [tex]\( y \)[/tex]. Multiply both sides by [tex]\( y^3 \)[/tex] to isolate the variable [tex]\( y \)[/tex] on one side:
[tex]\[ x y^3 = 19 \][/tex]
4. Isolate [tex]\( y^3 \)[/tex]. Divide both sides by [tex]\( x \)[/tex]:
[tex]\[ y^3 = \frac{19}{x} \][/tex]
5. Solve for [tex]\( y \)[/tex]. Take the cube root of both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[3]{\frac{19}{x}} \][/tex]
By solving for [tex]\( y \)[/tex], we obtain the inverse function:
[tex]\[ f^{-1}(x) = \sqrt[3]{\frac{19}{x}} \][/tex]
So, the correct inverse function is:
[tex]\[ f^{-1}(x) = \left( \frac{19}{x} \right)^{1/3} \][/tex]
Based on the given options, the correct answer is:
B. [tex]\( f^{-1}(x) = \left( \frac{19}{x} \right)^{1/3} \)[/tex]
1. Start with the function [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = \frac{19}{x^3} \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse. Set [tex]\( x \)[/tex] equal to the function and replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex]:
[tex]\[ x = \frac{19}{y^3} \][/tex]
3. Solve for [tex]\( y \)[/tex]. Multiply both sides by [tex]\( y^3 \)[/tex] to isolate the variable [tex]\( y \)[/tex] on one side:
[tex]\[ x y^3 = 19 \][/tex]
4. Isolate [tex]\( y^3 \)[/tex]. Divide both sides by [tex]\( x \)[/tex]:
[tex]\[ y^3 = \frac{19}{x} \][/tex]
5. Solve for [tex]\( y \)[/tex]. Take the cube root of both sides to isolate [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[3]{\frac{19}{x}} \][/tex]
By solving for [tex]\( y \)[/tex], we obtain the inverse function:
[tex]\[ f^{-1}(x) = \sqrt[3]{\frac{19}{x}} \][/tex]
So, the correct inverse function is:
[tex]\[ f^{-1}(x) = \left( \frac{19}{x} \right)^{1/3} \][/tex]
Based on the given options, the correct answer is:
B. [tex]\( f^{-1}(x) = \left( \frac{19}{x} \right)^{1/3} \)[/tex]
Answer:
[tex]y= \sqrt[3]{\frac{19}{x} }[/tex]
Step-by-step explanation:
To find the inverse of a function, follow these steps:
- Rewrite the equation so that f(x) = y: [tex]y=\frac{19}{x^{3} }[/tex]
- Switch the positions of x and y: [tex]x= \frac{19}{y^3}[/tex]
- Solve for y:
[tex]x= \frac{19}{y^3}[/tex]
- Multiply both sides by y³: [tex]y^3x=19[/tex]
- Divide both sides by x: [tex]y^3= \frac{19}{x}[/tex]
- Take the cubed root of both sides: [tex]y= \sqrt[3]{\frac{19}{x} }[/tex]
Your final answer is:
[tex]y= \sqrt[3]{\frac{19}{x} }[/tex]
Learn more about inverse functions here: https://brainly.com/question/30811064
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