Functions & Relations - Algebra 3/4

Card 0 of 9

Question

Find the composition, $f\circ g$ given

$\f(x)=x^2+2x \g(x)=3x$

Answer

To find the composition, $f\circ g$ given

$\f(x)=x^2+2x \g(x)=3x$

recall what a composition of two functions represents.

$f \circ g=f(g(x))$

This means that the function will replace each in the function .

Therefore, in this particular problem the composition becomes

$\f\circ g=f(g(x)) \f(g(x))=(3x)^2+2(3x) \f(g(x))=9x^2+6x$

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Question

Find the inverse of .

$f(x)=\frac{3}{2+5x}$

Answer

To find the inverse of a function swap the variables and solve for . The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function.

For this particular function the inverse is found as follows.

$f(x)=\frac{3}{2+5x}$

$y=\frac{3}{2+5x}$

First, switch the variables.

$x=\frac{3}{2+5y}$

Now, perform algebraic operations to solve for .

$x\cdot (2+5y)={3}$

$\frac{x\cdot(2+5y)}{x}=\frac{3}{x}$

$\2+5y=\frac{3}{x} \\5y=\frac{3}{x}-2 \\y=\frac{3}{5x}-\frac{2}{5}$

Therefore, the inverse is

$f^{-1}(x)=\frac{3}{5x}-\frac{2}{5}$

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Question

Find the composition, $f\circ g$ given

$\f(x)=2x^2+3 \g(x)=7x$

Answer

To find the composition, $f\circ g$ given

$\f(x)=2x^2+3 \g(x)=7x$

recall what a composition of two functions represents.

$f \circ g=f(g(x))$

This means that the function will replace each in the function .

Therefore, in this particular problem the composition becomes

$\f\circ g=f(g(x)) \f(g(x))=2(7x)^2+3 \f(g(x))=2(49x^2)+3\f(g(x))=98x^2+3$

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Question

Find the inverse of the function .

Answer

To find the inverse of a function swap the variables and solve for . The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function.

For this particular function the inverse is found as follows.

First, switch the variables.

Now, perform algebraic operations to solve for .

$\x-4=2y^2 \\\frac{x-4}{2}=y^2 \\\frac{x}{2}-2=y^2 \\\sqrt{\frac{x}{2}-2}=y$

Therefore, the inverse is

$f^{-1}(x)=\sqrt{\frac{x}{2}-2}$

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Question

Find the composition, $f\circ g$ given

$\f(x)=x^2 \\g(x)=\frac{1}{2}x$

Answer

To find the composition, $f\circ g$ given

$\f(x)=x^2 \\g(x)=\frac{1}{2}x$

recall what a composition of two functions represents.

$f \circ g=f(g(x))$

This means that the function will replace each in the function .

Therefore, in this particular problem the composition becomes

$\f\circ g=f(g(x)) \\f(g(x))=\left(\frac{1}{2}x \right )^2\\f(g(x))=\frac{1}{4}x^2$

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Question

Find the composition, $f\circ g$ given

$\f(x)=x^2+2x \g(x)=3x$

Answer

To find the composition, $f\circ g$ given

$\f(x)=x^2+2x \g(x)=3x$

recall what a composition of two functions represents.

$f \circ g=f(g(x))$

This means that the function will replace each in the function .

Therefore, in this particular problem the composition becomes

$\f\circ g=f(g(x)) \f(g(x))=(3x)^2+2(3x) \f(g(x))=9x^2+6x$

Compare your answer with the correct one above

Question

Find the inverse of .

$f(x)=\frac{3}{2+5x}$

Answer

To find the inverse of a function swap the variables and solve for . The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function.

For this particular function the inverse is found as follows.

$f(x)=\frac{3}{2+5x}$

$y=\frac{3}{2+5x}$

First, switch the variables.

$x=\frac{3}{2+5y}$

Now, perform algebraic operations to solve for .

$x\cdot (2+5y)={3}$

$\frac{x\cdot(2+5y)}{x}=\frac{3}{x}$

$\2+5y=\frac{3}{x} \\5y=\frac{3}{x}-2 \\y=\frac{3}{5x}-\frac{2}{5}$

Therefore, the inverse is

$f^{-1}(x)=\frac{3}{5x}-\frac{2}{5}$

Compare your answer with the correct one above

Question

Find the composition, $f\circ g$ given

$\f(x)=2x^2+3 \g(x)=7x$

Answer

To find the composition, $f\circ g$ given

$\f(x)=2x^2+3 \g(x)=7x$

recall what a composition of two functions represents.

$f \circ g=f(g(x))$

This means that the function will replace each in the function .

Therefore, in this particular problem the composition becomes

$\f\circ g=f(g(x)) \f(g(x))=2(7x)^2+3 \f(g(x))=2(49x^2)+3\f(g(x))=98x^2+3$

Compare your answer with the correct one above

Question

Find the inverse of the function .

Answer

To find the inverse of a function swap the variables and solve for . The function and its inverse when multiplied together, equals one. This means that the inverse undoes the function.

For this particular function the inverse is found as follows.

First, switch the variables.

Now, perform algebraic operations to solve for .

$\x-4=2y^2 \\\frac{x-4}{2}=y^2 \\\frac{x}{2}-2=y^2 \\\sqrt{\frac{x}{2}-2}=y$

Therefore, the inverse is

$f^{-1}(x)=\sqrt{\frac{x}{2}-2}$

Compare your answer with the correct one above

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