Composition of Functions - Pre-Calculus

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Suppose and

What would be?

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Substitute into the function for .

Then it will become:

What is ?

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f(g(x)) simply means: where ever you see an x in the equation f(x), replace it with g(x).

So, doing just that, we get

,

which simplifies to

.

Since

$\ln(e)=1$ our simplified expression becomes,

.

What is ?

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g(f(x)) simply means replacing every x in g(x) with f(x).

After simplifying, it becomes

$$\frac{3(2-5x)(2-5x)}{(2-5x)}$-6$

If , , and , what is ?

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When doing a composition of functions such as this one, you must always remember to start with the innermost parentheses and work backward towards the outside.

So, to begin, we have

.

Now we move outward, getting

.

Finally, we move outward one more time, getting

.

For the functions

and

.

Evaluate the composite function

$f\circ g$ .

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The composite function means to plug in the function of into the function for every x value in the function.

Therefore the composition function becomes:

$f\circ g = f[g(x)] = $4(x^2$) = $4x^2$$ .

For the functions

and

.

Evaluate the composite function

$f\circ g$ .

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The composite function means to plug in the function into for every x value.

Therefore the composite function becomes,

$f\circ g = f[g(x)] = $(x^2$)+4 = $x^2$+4$

Find if , , and .

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Solve for the value of .

Solve for the value of .

Solve for the value .

Let

Determine $f\circ g :(4)$ .

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To find the composite function we start from the most inner portion of the expression and work our way out.

$f\circ g :(4) = $f[g(4)]=f(0)=e^0$=1$

Let

Determine

$f\circ g$ .

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The composite funtion means to replace every entry x in f(x) with the entire function g(x).

$f\circ g= $f[g(x)]=(x^2$$)+4=x^2$+4$

For the functions and , evaluate the composite function $f\circ g$

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The composite function notation $f\circ g$ means to swap the function into for every value of . Therefore:

$f\circ g$

For the functions and , evaluate the composite function $f\circ g$ .

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The composite function notation $f\circ g$ means to swap the function into for every value of . Therefore:

$f\circ g$

For the functions and , evaluate the composite function $g\circ f$ .

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The composite function notation $g\circ f$ means to swap the function into for every value of . Therefore:

$g\circ f$

For , $g(x)=\frac{x}{3}$$ , and $h(x)=3$\sqrt{x+1}$$ , determine .

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Working inside out, first do .

This is,

$\frac{3 \sqrt{x+1}$}{3}= $\sqrt{x+1}$ .

Now we will do $f($\sqrt{x+1}$)$ .

This is

For $k(x)=ln(x), $f(x)=e^{2x}$, $j(x)=x^2$-3$ , write a function for .

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Working from the inside out, first we will find a function for .

This is:

, which we can simplify slightly to .

Now we will plug this new function into the function k:

.

Since ln is the inverse of e to any power, this simplifies to .

Find $g\cdot f$ given the following equations

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To find $g\cdot f$ simply substiute for every x in and solve.

We are given the following:

and .

Find:

$(f\circ g)(x)$

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Let's discuss what the problem is asking us to solve. The expression $(f\cdot g)(x)$ (read as as "f of g of x") is the same as . In other words, we need to substitute into .

Substitute the equation of for the variable in the given function:

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

Next we need to FOIL the squared term and simplify:

FOIL means that we multiply terms in the following order: first, outer, inner, and last.

First: $x\cdot $x=x^2$$

Outer: $x\cdot 3=3x$

Inner: $3\cdot x=3x$

Last: $3\cdot 3=9$

When we combine like terms, we get the following:

Substitute this back into the equation and continue to simplify.

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

$(f\cdot $g)(x)=x^2$+6x+12$

None of the answers are correct.

Find $(g\cdot f)(x)$ and evaluate at .

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$(g\cdot f)(x)$

$g(x)={\color{Red} $x^2$-x}$

"G of F of X" means substitute f(x) for the variable in g(x).

$(g\cdot f)(x)=(x-5){\color{Red} ^2}-(x-5)$

Foil the squared term and simplify:

First: $x\cdot $x=x^2$$

Outer: $x\cdot -5=-5x$

Inner: $-5 \cdot x= -5x$

Last: $-5 \cdot -5=25$

$(g\cdot $f)(x)=x^2$$-10x+25-x+5=x^2$-11x+30$

So $(g\cdot f)(x)={\color{Blue} $x^2$-11x+30}$

Now evaluate the composite function at the indicated value of x:

$(g\cdot $f)(5)=5^2$-11(5)+30={\color{Blue} 0}$

Find if and .

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Replace and substitute the value of into so that we are finding .

Given and , find .

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Given and , find .

Begin by breaking this into steps. I will begin by computing the step, because that will make the late steps much more manageable.

Next, take our answer to and plug it into .

So we are close to our final answer, but we still need to multiply by 3.

Making our answer 84.

and . Find $(f\circ g)(x)$ .

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${\color{Blue} $f(x)=x^2$-3}$ and ${\color{Red} g(x)=2x+3}$ .

To find $(f\circ g)(x)$ we plug in the function everywhere there is a variable in the function .

This is the composition of "f of g of x".

$(f\circ g)(x)=f(g(x))={\color{Blue} (}{\color{Red} 2x+3}{\color{Blue} )}{\color{Blue} ^2-3}$

Foil the square and simplify:

$(f\circ $g)(x)=(4x^2$$+12x+9)=\mathbf{4x^2$+12x+6}$