Algebra 3/4 - Algebra 3/4

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Question

What is the degree equivalent for an angle that is $\frac{\pi}{2}$ radians?

Answer

To find the degree equivalent for an angle that is $\frac{\pi}{2}$ radians first recall the unit conversion between degrees and radians.

There are 360 degrees or $2\pi$ radians in a circle.

When simplified this is,

$\pi=180^\circ$

therefore to convert from radians to degrees, simply multiply by $\frac{180^\circ}{\pi}$ .

$\frac{\pi}{2}\times \frac{180^\circ}{\pi}$

$\frac{180^\circ}{2}=90^\circ$

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Question

What is the degree equivalent for an angle that is $\frac{3\pi}{2}$ radians?

Answer

To find the degree equivalent for an angle that is $\frac{\pi}{2}$ radians first recall the unit conversion between degrees and radians.

There are 360 degrees or $2\pi$ radians in a circle.

When simplified this is,

$\pi=180^\circ$

therefore to convert from radians to degrees, simply multiply by $\frac{180^\circ}{\pi}$ .

$\frac{3\pi}{2}\times \frac{180^\circ}{\pi}$

$\frac{3\cdot 180^\circ}{2}=270^\circ$

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Question

What is the degree equivalent for an angle that is $\frac{\pi}{2}$ radians?

Answer

To find the degree equivalent for an angle that is $\frac{\pi}{2}$ radians first recall the unit conversion between degrees and radians.

There are 360 degrees or $2\pi$ radians in a circle.

When simplified this is,

$\pi=180^\circ$

therefore to convert from radians to degrees, simply multiply by $\frac{180^\circ}{\pi}$ .

$\frac{\pi}{2}\times \frac{180^\circ}{\pi}$

$\frac{180^\circ}{2}=90^\circ$

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Question

What is the degree equivalent for an angle that is $\frac{3\pi}{2}$ radians?

Answer

To find the degree equivalent for an angle that is $\frac{\pi}{2}$ radians first recall the unit conversion between degrees and radians.

There are 360 degrees or $2\pi$ radians in a circle.

When simplified this is,

$\pi=180^\circ$

therefore to convert from radians to degrees, simply multiply by $\frac{180^\circ}{\pi}$ .

$\frac{3\pi}{2}\times \frac{180^\circ}{\pi}$

$\frac{3\cdot 180^\circ}{2}=270^\circ$

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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

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

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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

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

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Question

Convert the expression from logarithmic to exponential.

$\log_6 216=3$

Answer

To convert the logarithmic expression to exponential form, recall the change of base formula.

$\log_b a=c\rightarrow b^c=a$

Apply the change of base formula to this particular logarithmic expression.

$\log_6 216=3$

$\b=6 \a=216 \c=3$

The exponential form becomes

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Question

Convert the expression from logarithmic to exponential.

$\log_4 1024=5$

Answer

To convert the logarithmic expression to exponential form, recall the change of base formula.

$\log_b a=c\rightarrow b^c=a$

Apply the change of base formula to this particular logarithmic expression.

$\log_4 1024=5$

$\b=4 \a=1024 \c=5$

The exponential form becomes

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Question

Convert the expression from logarithmic to exponential.

$\log_3 9=2$

Answer

To convert the logarithmic expression to exponential form, recall the change of base formula.

$\log_b a=c\rightarrow b^c=a$

Apply the change of base formula to this particular logarithmic expression.

$\log_3 9=2$

$\b=3 \a=9 \c=2$

The exponential form becomes

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Question

Convert the expression from logarithmic to exponential.

$\log_6 216=3$

Answer

To convert the logarithmic expression to exponential form, recall the change of base formula.

$\log_b a=c\rightarrow b^c=a$

Apply the change of base formula to this particular logarithmic expression.

$\log_6 216=3$

$\b=6 \a=216 \c=3$

The exponential form becomes

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Question

Convert the expression from logarithmic to exponential.

$\log_4 1024=5$

Answer

To convert the logarithmic expression to exponential form, recall the change of base formula.

$\log_b a=c\rightarrow b^c=a$

Apply the change of base formula to this particular logarithmic expression.

$\log_4 1024=5$

$\b=4 \a=1024 \c=5$

The exponential form becomes

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Question

Simplify the following radical expression.

$\sqrt{125}+\sqrt{48}$

Answer

To simplify the radical expression look at the factors under each radical.

$\sqrt{125}+\sqrt{48}$

$\sqrt{25\cdot 5}+\sqrt{4\cdot 12}$

Recall that 25 and 4 are perfect squares.

$5\sqrt{5}+2\sqrt{12}$

From here 12 can be factored further.

$\5\sqrt{5}+2\sqrt{4\cdot 3} \5\sqrt{5}+2\cdot 2\sqrt{3} \5\sqrt{5}+4\sqrt{3}$

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Question

Simplify the following radical expression.

$\sqrt{512}+\sqrt{68}$

Answer

To simplify the radical expression look at the factors under each radical.

$\sqrt{512}+\sqrt{68}$

$\sqrt{16\cdot 16\cdot 2}+\sqrt{4\cdot 17}$

Recall that 16 and 4 are perfect squares.

$16\sqrt{2}+2\sqrt{17}$

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