Composition of Functions - Pre-Calculus

Card 0 of 116

Question

Suppose and

What would be?

Answer

Substitute into the function for .

Then it will become:

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Question

$f(x)=3-2x^2+\ln(x)$

What is ?

Answer

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

$3-2(e^x)^2+\ln(e^x)$ ,

which simplifies to

$3-2e^{2x}+x\ln(e)$ .

Since

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

$3-2e^{2x} +x$ .

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Question

$g(x)=\frac{3x^2}{x}-6$

What is ?

Answer

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

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

After simplifying, it becomes

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

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Question

If , , and , what is ?

Answer

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

.

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Question

For the functions

and

.

Evaluate the composite function

$f\circ g$ .

Answer

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:

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

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Question

For the functions

and

.

Evaluate the composite function

$f\circ g$ .

Answer

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

Therefore the composite function becomes,

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

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Question

Find if , , and .

Answer

Solve for the value of .

Solve for the value of .

Solve for the value .

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Question

Let

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

Answer

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

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

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Question

Let

Determine

$f\circ g$ .

Answer

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

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

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Question

For the functions and $g(x)=x^{3}$ , evaluate the composite function $f\circ g$

Answer

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

$f\circ g$

$=(x^{3})+7$

$=x^{3}+7$

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Question

For the functions $f(x)=x^{2}-4$ and , evaluate the composite function $f\circ g$ .

Answer

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

$f\circ g$

$=(2x+3)^{2}-4$

$=4x^{2}+12x+9-4$

$=4x^{2}+12x+5$

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Question

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

Answer

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

$g\circ f$

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Question

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

Answer

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 $(\sqrt{x+1})^2 = x+1$

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Question

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

Answer

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

This is:

$e^{2(x^2 -3)}$ , which we can simplify slightly to $e^{2x^2 -6}$ .

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

$ln(e^{2x^2-6})$ .

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

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Question

Find $g\cdot f$ given the following equations

Answer

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

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Question

If and , find $(f\circ g)(x)$ .

Answer

First, make sure that g $\subseteq$ f (range of g is a subset of the domain of f).

Since the g: $\left { y|-1\leq y\leq 1\right }$ and f: $\left { x|x\epsilon \mathbb{R}\right }$ , g $\subseteq$ f and $(f\circ g)(x)$ exists.

Plug in the output of , which is $\sin (x)$ , as the input of .

Thus,

$(f\circ g)(x)=1-(\sin (x))^2=1-\sin^2(x)=\cos^2 (x)$

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Question

We are given the following:

$f(x)=x^2+3\hspace{5mm}$ and .

Find:

$(f\circ g)(x)$

Answer

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.

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Question

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

Answer

$(g\cdot f)(x)$

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

"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:

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

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Question

Find if and .

Answer

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

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Question

Given and , find .

Answer

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.

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