Chapter 3: Q. 80 (page 157)

In Problems 75–80, for the given functions $f$ and $g .$
(a) Graph $f$ and $g$ on the same Cartesian plane.
(b) Solve $f (x) = g (x) .$
(c) Use the result of part (b) to label the points of intersection of the graphs of $f$ and $g .$
(d) Shade the region for which $f (x) > g (x),$ that is, the region below $f$ and above $g$
$f (x) = - x^{2} + 7 x - 6; g (x) = x^{2} + x - 6$

Short Answer

Expert verified

(a) The graph of $f$ and $g .$

(b) $x = 0, 3 .$

(d) The shaded region

Step by step solution

Part (a) Step 1. Given information

It is given that $f (x) = - x^{2} + 7 x - 6; g (x) = x^{2} + x - 6 .$ We need to draw the graph of $f$ and $g$ in the same cartesian plane.

Step 2. Draw the graph

Part b Step 1. Given information

It is given that $f (x) = - x^{2} + 7 x - 6; g (x) = x^{2} + x - 6 .$ We need to solve $f (x) = g (x) .$

Step 2. Solving

$f (x) = g (x) .$

$- x^{2} + 7 x - 6 = x^{2} + x - 6 .$

$2 x^{2} - 6 x = 0 .$

$x (x - 3) = 0 .$

$x = 0, a n d x - 3 = 0 .$

$x = 3 .$

The values of the function $f$ and $g$ are equal, when $x = 0, 3 .$

Part (c) Step 1. Given information

It is given that $f (x) = - x^{2} + 7 x - 6; g (x) = x^{2} + x - 6 .$ We need to use the result of part (b) to label the point of intersection.

Step 2. Simplify

Let $y = f (x) = - x^{2} + 7 x - 6 .$

$y = g (x) = x^{2} + x - 6 .$

When $x = 0, y = 0^{2} + 0 - 6 = - 6 .$

When $x = 3, y = 3^{2} + 3 - 6 = 6 .$

The point of intersection is $(0, - 6), (3, 6) .$

Part (d) Step 1. Given information

It is given that $f (x) = - x^{2} + 7 x - 6; g (x) = x^{2} + x - 6 .$ We need to shade the region for which $f (x) > g (x) .$

Step 2. Shade the region

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x	$y = f (x)$
-2	-4
-1	0
0	4
1	8
2	12

Short Answer

Step by step solution

Part (a) Step 1. Given information

Step 2. Draw the graph

Part b Step 1. Given information

Step 2. Solving

Part (c) Step 1. Given information

Step 2. Simplify

Part (d) Step 1. Given information

Step 2. Shade the region

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In Problems 75–80, for the given functions fand g.(a) Graph fand gon the same Cartesian plane.(b) Solve fx=gx.(c) Use the result of part (b) to label the points of intersection of the graphs of fand g.(d) Shade the region for which fx>gx,that is, the region below fand above gfx=-x2+7x-6;gx=x2+x-6

Short Answer

Step by step solution

Part (a) Step 1. Given information

Step 2. Draw the graph

Part b Step 1. Given information

Step 2. Solving

Part (c) Step 1. Given information

Step 2. Simplify

Part (d) Step 1. Given information

Step 2. Shade the region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Geometry

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.