Respuesta :
Answer:
[tex]f(4)=e^{4}-3[/tex] or [tex]f(4)\approx51.5981500331...[/tex]
Step-by-step explanation:
The following is a general explanation of functions, what they mean, and how to evaluate them. The simplifications and calculations at the end are based on the clarification from your comments that [tex]f(x)=e^{x}-3[/tex] . If [tex]f(x)[/tex] is equal to something else, the simplifications/calculation will obviously be a little different, and the answer will likely be different, but the general process will be the same.
Understanding function notation
In the equation [tex]f(x)=e^{x}-3[/tex] , the left side of the equation is saying several things at once.
Names
The first character, "f", is the name of the function.
The name of a function tells you which set of directions you should look for to apply to an input. Some functions have names that are more than one letter, and even might give a hint as to what they do, for instance I might name a certain function [tex]squareroot[/tex], and based on the name, you might guess what the equation for [tex]squareroot(x)[/tex] might look like.
Mathematicians often use "f" as a function name between different problems over and over and over again, unless they come up with a special function that is worth giving a special name to.
Input
The rest of the characters after the name of the function, are "(x)" which is a pair of parentheses to contain the input, and the input itself.
Again, mathematicians often use "x" as a generic input over and over and over again, until we actually know which number we want to put into the function.
Output
Even though we've already discussed every character in the notation on the left, there is actually one more thing that the left side of the equation is saying: Altogether, "f(x)" is the output of the function, when the input "x" is put in. I feel it's worth stating, because it's often glossed over.
The equation itself
Next in the equation is the equals sign, which means that the left side of the equation, which is the output "f(x)", is equal to the stuff on the right.
The right side of the equation is the rule to follow describing what f(x) does to an input, and can help you calculate the output.
For instance, if I asked what p(4) was, it is clear that there is an input of 4, but without knowing what the function "p" does, there's no way to know what the function will give as an output when one inputs "4". This is why the right-hand side of the equation for a function is important.
Evaluating functions
To evaluate any function, we take an input, apply the function, and simplify/calculate the output.
In this case, the question asks for [tex]f(4)[/tex], meaning that the "input" is "4".
In this case, the question gives an explanation of what the function does through a rule given in equation form.
Given your clarification of the given function, [tex]f(x)=e^{x}-3[/tex] , meaning a generic input (in this case "x") is put into the function "f" and the output is found by doing the arithmetic on the right side of the equation (exponentiating the number by "e", and then subtracting 4)
Notice that in the expression [tex]f(4)[/tex], the "x" was replaced with "4". So, we need to substitute "4" where we see an "x" in the rule on the right hand side of the equation rule, and simplify and/or calculate (calculate, like with a calculator)
So, to evaluate the expression [tex]f(4)[/tex], simply substitute, and simplify/calculate.
Simplifying (no calculator)
To simplify without a calculator, we substitute "4". Any time you make a substitution, use parentheses. Then, try to simplify.
[tex]f(4)=e^{(4)}-3[/tex]
[tex]f(4)=e^{4}-3[/tex]
As it turns out, this is as much as we could simplify things here (without a calculator). So, if an exact answer is needed, this would be it.
Calculating (with a calculator)
To calculate with a calculator, we substitute "4". Any time you make a substitution, use parentheses. Then, try to simplify.
[tex]f(4)=e^{(4)}-3[/tex]
[tex]f(4)=e^{4}-3[/tex]
[tex]f(4)=54.5981500331...-3[/tex]
[tex]f(4)=51.5981500331...[/tex]
So calculating with a calculator, the output of the function "f", for an input of "4", is approximately 51.598.
