Use the given functions f and g f(x)=-x^2+3 g(x)=-3 x+3

Name: Problem 26, Chapter 4: Algebra and Trigonometry
Uploaded: 2020-07-06T14:56:03-08:00
Duration: 6 min 40 s
Channel: Ethan Dotson-Kelly
Description: Use the given functions f and g f(x)=-x^2+3 g(x)=-3 x+3

Use the given functions f and g f(x)=-x^2+3 g(x)=-3...

Key Concepts

Function Composition and Operations

This concept covers combining functions in various ways, such as through composition, addition, subtraction, multiplication, or division. It involves applying one function to the result of another, which is a common operation when working with multiple functions.

Linear Functions

Linear functions are first-degree functions represented in slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept. They produce straight-line graphs and are fundamental in analyzing rates of change and relationships between variables.

Function Notation

This concept involves the use of symbols to represent functions, where an input value is related to an output value through a defined rule. Understanding function notation is essential for evaluating functions at specific points, composing functions, and performing various operations on them.

Quadratic Functions

Quadratic functions are polynomials of degree two that graph as parabolas. They are typically represented in the form ax² + bx + c. Key features include the vertex, axis of symmetry, and direction of opening, which are crucial for understanding the graph and behavior of the function.

Transcript

00:01 This problem is asking us to solve six different questions given our problems over here on the right.

00:07 So we have a function f of x and f of x is equal to negative x squared plus three.

00:12 And knowing what i know about functions, the x squared makes this a parabola.

00:18 And as a reminder, when it comes a function notation, x is my input value, while f of x is my output value.

00:25 Same thing here, g of x.

00:27 So x is my input and g of x is my output.

00:31 And so we're looking at two different situations here.

00:35 One with a parabola, one with a linear line, and we're looking at all six of these problems.

00:41 And when you look at these problems, we need to think to ourselves, what does it mean for f of x equals zero? well, we know that x is our input value, but what is equalling zero right now? the f of x is equaling zero, which means my output value is zero.

00:57 Know our output value, the question here is, what is my input value? one of the easiest ways to determine that is to look at a graph.

01:06 So coming over here, i am looking for when is the output value equal to zero? well, that's at the x -axis.

01:12 So let's take a look at the graph.

01:14 The graph is here.

01:15 F of x equals x squared.

01:18 Let's go ahead and zoom this back into the main area.

01:21 And you'll see that it crosses the x -axis at two locations, negative 1 .732 and positive.

01:27 1 .732.

01:29 And again, the output value, the y value, is zero at these two locations.

01:35 So 1 .732 is my answer, both positive and negative.

01:39 So i can say x equals negative 1 .732 or x equals 1 .732.

01:55 The same thing is being asked for g of x now for question b.

01:59 For question.

02:00 We are asking when is the input value? what is the input value when the output value is equal to zero? so coming back over here, again i have the graph already made for us, negative 3x plus 3.

02:13 And looking at how it's drawn here, it crosses the x -axis at the point 1 -0.

02:19 So that's going to be my answer.

02:21 X equals 1 when the output value is 0.

02:26 Question c, when does f of x equal g of x.

02:33 And again, we're asking the output values here...

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