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(a) determine the domain, (b) sketch the graph and (c) determine the range of the function defined by the given equation.$g(x)=x^{2}+5$

(a) $-\infty<x<\infty$(b)(c) $y \geq 5$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

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01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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(a) determine the domain, …

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(a) find the domain of the…

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Sketch a graph of each fun…

for this problem. We've been given a function g of X equals X squared plus five. For this function, we're going to find three things. We want to find the domain and the range from our function. Plus, we want to sketch it so we can see what we have. Sometimes visualizing the function helps us to see the domain and range a little more clearly. So domain and range first for my domain. Those air my ex values all the input sick and go into my function. And when we're doing domain, we're looking at valid exes. It's often useful to look at it in reverse. Are there any excess that are not possible? Well, X squared plus five. I can square any number. Any real number could be squared. Zero negative. Positive. Um, pie, You pick the number, I can square it. So the domain has nothing, has nothing that we can't put in it. So it includes all real numbers. So we can write this in two ways. I can say that X is between negative infinity and positive infinity, or we can write it in interval notation. In either case, it means we have all real numbers included. What about the range, though? Range or wise? Those are my outputs. Well, if I look at this function, X squared is always zero or positive. It can never be negative. So the very, very smallest number the G of X ca NBI is five. I can't ever be smaller than five because x squared at best, I'm gonna be adding zero. But usually I'm gonna be adding another positive number 25 So why has to be greater than or equal to five? That's my range. I could get wise, Bigas I want. But I could never be smaller than five If we want to write that in interval notation, that would be negative. Infinity to five and it includes five because of X zero. My function is five. So let's graph this. This is the graph of a parabola, and the best way to do this is often just to pick some points. We already saw that X equaling zero. Give me a Y of five, and I'm always gonna be above that point. Nothing. Nothing down here. So let's just pick a number. Let's let X equal one. Well, that means I've got why it's six and negative one is gonna be the same when you square it won a negative one or the same. What if excess too? Well, that means I'll have five plus four, which is nine, 6789 And again, that'll be symmetric over here. So you can see as I connect these points, there's my parabola. Visually, you could tell that I have nothing under Y equals five, but my exes. I can keep going in either direction, positive or negative. So my domain will include all real numbers, so sketch domain and range for my given function.

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