Chapter 1: Q. 1 (page 119)

Finding roots of piecewise-defined functions: For each function f that follows, find all values x = c for which f(c) = 0. Check your answers by sketching a graph of f.
$\begin{aligned} f (x) = \{\begin{aligned} 4 - x^{2}, & if x < 0 \\ x + 1, & if x \geq 0 \end{aligned} \\ f (x) = \{\begin{aligned} x + 1, & if x < 0 \\ 4 - x^{2}, & if x \geq 0 \end{aligned} \\ f (x) = \{\begin{aligned} 2 x - 1, & if x \leq 1 \\ 2 x^{2} + x - 3, & if x > 1 \end{aligned} \end{aligned}$

Short Answer

Expert verified

$f (x) = \{\begin{aligned} 4 - x^{2}, & if x < 0 \\ x + 1, & if x \geq 0 \end{aligned}$ has no roots.

$f (x) = \{\begin{aligned} x + 1, & if x < 0 \\ 4 - x^{2}, & if x \geq 0 \end{aligned}$ we have x=-1,2

$f (x) = \{\begin{aligned} 2 x - 1, & if x \leq 1 \\ 2 x^{2} + x - 3, & if x > 1 \end{aligned}$ we have $x = 1, \frac{1}{2}$

Step by step solution

Step 1. Given information

We have to find the roots of the following functions :

$\begin{aligned} f (x) = \{\begin{aligned} 4 - x^{2}, & if x < 0 \\ x + 1, & if x \geq 0 \end{aligned} \\ f (x) = \{\begin{aligned} x + 1, & if x < 0 \\ 4 - x^{2}, & if x \geq 0 \end{aligned} \\ f (x) = \{\begin{aligned} 2 x - 1, & if x \leq 1 \\ 2 x^{2} + x - 3, & if x > 1 \end{aligned} \end{aligned}$

Step 2. Finding roots.

$\begin{aligned} let 4 - x^{2} = 0 (x < 0) \\ \begin{matrix} x = - 2 \\ let x + 1 = 0 (x \geq 0) \\ x = - 1 \end{matrix} \end{aligned}$

It can not be considered as root.

$f (x) = \{\begin{cases} x + 1 : x < 0 \\ 4 - x^{2} : x \geq 0 \end{cases}$

Let x+1=0

x=-1

Let $4 - x^{2} = 0 x = + 2$

$f (x) = \{\begin{cases} 2 x - 1 : x \leq 1 \\ 2 x^{2} + x - 3 : x > 1 \end{cases}$

Let 2x-1=0

$x = \frac{1}{2}$

Let role="math" localid="1649831777557" $2 x^{2} + x - 3 = 0 (x > 1) x = 1, \frac{1}{2}$

No roots.

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding roots.

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Finding roots of piecewise-defined functions: For each function f that follows, find all values x = c for which f(c) = 0. Check your answers by sketching a graph of f.f(x)=4−x2,ifx<0x+1,ifx≥0f(x)=x+1,ifx<04−x2,ifx≥0f(x)=2x−1,ifx≤12x2+x−3,ifx>1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding roots.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Logic and Functions

Pure Maths

Mechanics Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.