If f(x)=x+5 and g(x)=x^2-3, find the following. a. f(g(0)) b. g(f(0)) c. f(g(x))

If f(x)=x+5 and g(x)=x^2-3, find the following. a. ...

If $f(x)=x+5$ and $g(x)=x^{2}-3,$ find the following. $$ \begin{array}{ll}{\text { a. } f(g(0))} & {\text { b. } g(f(0))} \\ {\text { c. }} {f(g(x))} & {\text { d. } g(f(x))} \\ {\text { e. }} {f(f(-5))} & {\text { f. } g(g(2))} \\ {\text { g. }} {f(f(x))} & {\text { h. } g(g(x))}\end{array} $$

If $f(x)=x+5$ and $g(x)=x^{2}-3,$ find the following.
$$
\begin{array}{ll}{\text { a. } f(g(0))} & {\text { b. } g(f(0))} \\ {\text { c. }} {f(g(x))} & {\text { d. } g(f(x))} \\ {\text { e. }} {f(f(-5))} & {\text { f. } g(g(2))} \\ {\text { g. }} {f(f(x))} & {\text { h. } g(g(x))}\end{array}
$$

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Key Concepts

Function Composition

Function composition involves applying one function to the result of another function. It is denoted by f(g(x)) where the function g is evaluated first, and its output is passed as the input to f. Understanding the order and process of this operation is crucial in manipulating and analyzing functions.

Function Evaluation

Function evaluation is the process of substituting a particular input value into a function and computing the corresponding output. This fundamental concept is used to determine the value of a function for specific arguments.

Algebraic Manipulation

Algebraic manipulation includes the skills of substituting expressions, simplifying algebraic expressions, and combining like terms. These skills are essential when working with functions, especially during the composition and evaluation processes.

Transcript

00:01 For this problem, we're given f of x equal to x plus 5 and g of x equal to x squared minus 3.

00:06 And we are to find the following composition of functions.

00:11 For the first one, we have f of g of 0.

00:17 And to evaluate this first, we need to find the value of g when x is 0.

00:25 That'll be f, and then we have 0 squared minus 3.

00:33 That's f of negative 3 and then you plug in negative 3 into f we get negative 3 plus 5 equal to 2 for b we want to find g of f of 0 this time we first want to find the value of f when x is 0 so that's 0 plus 5 that's g of 5 and then we go back to g and plug in 5 for x that'll be 5 5 5 that'll be 5 5 5 5 5.

01:06 Squared minus 3 or 25 minus 3 equal to 22.

01:14 Then for part c, we have f of g of x.

01:18 Define f of g of x, we replace every x in f by g of x.

01:27 So from an x plus 5, it becomes x squared minus 3 plus 5.

01:33 That's x squared minus 3 plus 5 or plus 2.

01:38 And then for g of f of x, we replace every f or every x in g by f of x.

01:47 So from an x squared minus 3, it becomes x plus 5 squared minus 3.

01:54 That's x squared plus 10x plus 25 minus 3, which equals x squared plus 10x plus 22.

02:06 Next, we want to find f of f of negative 5.

02:10 So every x in f gets replaced by negative 5...