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Figure $5(a)$ shows the graph of a piecewise linear function $f$ Find the algebraic formulation of $f(x) .$ Find $f(-2), f(0), f(2),$ and $f(5)$

f(x)=\left\{\begin{array}{cc}\frac{-7 x-3}{2} & x<1 \\2 & 1 \leq x<3 f(-2)=11 / 2, f(0)=-3 / 2, f(2) \\6 x-22 & x \geq 3\end{array}\right.

Algebra

Chapter 1

Functions and their Applications

Section 3

Applications of Linear Functions

Functions

University of Michigan - Ann Arbor

Idaho State University

Lectures

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).

03:18

03:47

Figure $5(\mathrm{c})$ sho…

Figure $5(b)$ shows the gr…

00:35

Find the slope of the grap…

01:25

Write the linear function …

02:20

Writing a Linear Function.…

Graphing Linear Functions …

using piece wise functions. We're gonna be determining the function of ffx as from the given information we have in the graph of FX here to begin, let's start by finding each of the equations for Lines A, B and C To do that, let's find Slope by recalling the information and blue listed at top. That slope is equal. So why? To minus y one over x two minus x one. Plugging in our coordinate points of line A. We find that slope is equal to negative 7/2 doing the same for Line B. We find that we have a slope equal to two, and for see, we find that we have a slope equal to six. Now we can plug these slope values and one of our coordinate points for each line into Point Slope form in order to determine the functions for each one of them individually doing this, we find that for a we have a function which is equal to negative seven X minus three all over to for B. We find that we have a function which is why equal to two and see we have a function equal to why equal to six x minus 22. You can double check that work by plugging your values into point slope. Form yourself and you'll arrive at the same conclusion. Now we want to determine for which values of X do these equations apply. To do that, we take a look at our graph. So for a we see that we have a closed circle. Sitting at X is equal to negative one. That means that it can include negative one, but we also come, but it extends beyond that as well. So it goes into infinity there. But we see that at 0.1 negative. Five, we have an open circle, meaning that it cannot include one. So for any values that are less than one, it looks like this will apply. So for our values, X less than one we can use the equation a function A. As for B, we can see that it sits between one and three for X. We have a one right here, three thio here, however, it can include one, so it's going to be greater than or equal to one. But it's got to be less than three for C. We see that for value any values greater than or equal to three. This will apply because it extends beyond up here, starting at three. Down to here. So using these were able to determine that we have a piece wise function, so that looks something like this. So f of X is equal to total of three functions. We have negative seven X minus three, divided by two And that's if X is less center. Less than one F of X is equal to if X is greater than or equal to one, but less than three. And that f of X is equal to six x minus 22 if X is greater than or equal to three. Now suppose we wanna find these values if X is equal to negative two. To do that, we see that negative two is less than negative one. Meaning we're gonna be using our equation A for this plugging negative to into that equation, we have negative seven times negative two minus three, divided by two tells us that at if X is equal to negative two, we have a value of 11 over to. If F is equal to zero again, we're gonna be using our function. A because zero is less than one plugging that in. We have negative seven times zero minus three all over to giving us a value of negative three over to if X is equal to zero. If X is equal to two, we're gonna be using our second equation because that's why equals to it falls within our bounds off between one and less than three. We have just equal to to that that's an easy one. And if F is equal to five or X is equal to five, we're gonna be using our equations. See, because five is greater than or equal to three, it falls within those bounds. Doing this. We have six times five minus 22 which is equal to eight, meaning that when X is equal to five f of X is equal to eight.

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