Let g(x)=5-e^(x). Below is Sean's attempt to write a formal justification for the fact that the equation g(x)=0 has a solution where 1 <= x <= 4. Is Sean's justification complete? If not, why? Sean's justification: g is defined for all real numbers, and exponential functions are continuous at all points in their domains. So, according to the intermediate value theorem, g(x)=0 must have a solution somewhere between x=1 and x=4. Choose 1 answer: (A) Yes, Sean's justification is complete. (B) No, Sean didn't establish that 0 is between g(1) and g(4). (C) No, Sean didn't establish that g is continuous.

Evaluate g(x): Let's first evaluate g(x) at x = 1 and x = 4 to determine the values of g(1) and g(4). g(1) = 5 - e^(1) = 5 - e g(4) = 5 - e^(4)Check Straddling of 0: Now we need to check if g(1) and g(4) straddle 0, which means one should be positive and the other should be negative for the Intermediate Value Theorem to guarantee a solution for g(x) = 0 between x = 1 and x = 4. Since e is approximately 2.71828, we can see that g(1) = 5 - e is positive because 5 is greater than e.Evaluate g(4): Next, we need to evaluate whether g(4) is less than 0. g(4) = 5 - e^(4), and since e^(4) is much larger than 5, g(4) will be negative.Apply Intermediate Value Theorem: Since g(1) is positive and g(4) is negative, and g(x) is continuous (as it is a combination of continuous functions: a constant, subtraction, and an exponential function), the Intermediate Value Theorem applies.Incomplete Justification: Sean's justification is incomplete because he did not explicitly establish that g(1) is greater than 0 and g(4) is less than 0, which is necessary to apply the Intermediate Value Theorem to ensure that a zero of g(x) exists between x = 1 and x = 4.

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Let
g(x)=5-e^(x).
Below is Sean's attempt to write a formal justification for the fact that the equation
g(x)=0 has a solution where
1 <= x <= 4.
Is Sean's justification complete? If not, why?
Sean's justification:

g is defined for all real numbers, and exponential functions are continuous at all points in their domains.
So, according to the intermediate value theorem,
g(x)=0 must have a solution somewhere between
x=1 and
x=4.
Choose 1 answer:
(A) Yes, Sean's justification is complete.
(B) No, Sean didn't establish that 0 is between
g(1) and
g(4).
(C) No, Sean didn't establish that
g is continuous.

Let $g(x)=5-e^{x}$ . $\newline$ Below is Sean's attempt to write a formal justification for the fact that the equation $g(x)=0$ has a solution where $1 \leq x \leq 4$ . $\newline$ Is Sean's justification complete? If not, why? $\newline$ Sean's justification: $\newline$ $g$ is defined for all real numbers, and exponential functions are continuous at all points in their domains. $\newline$ So, according to the intermediate value theorem, $g(x)=0$ must have a solution somewhere between $x=1$ and $x=4$ . $\newline$ Choose $1$ answer: $\newline$ (A) Yes, Sean's justification is complete. $\newline$ (B) No, Sean didn't establish that $0$ is between $g(1)$ and $g(4)$ . $\newline$ (C) No, Sean didn't establish that $g$ is continuous.

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Math Problems

Grade 6

One-step inequalities: word problems

Full solution

Q. Let

g(x)=5-e^{x}

\newline

Below is Sean's attempt to write a formal justification for the fact that the equation

g(x)=0

has a solution where

1 \leq x \leq 4

\newline

Is Sean's justification complete? If not, why?

\newline

Sean's justification:

\newline

g

is defined for all real numbers, and exponential functions are continuous at all points in their domains.

\newline

So, according to the intermediate value theorem,

g(x)=0

must have a solution somewhere between

x=1

and

x=4

\newline

Choose

1

answer:

\newline

(A) Yes, Sean's justification is complete.

\newline

(B) No, Sean didn't establish that

0

is between

g(1)

and

g(4)

\newline

g

is continuous.

Evaluate $g(x)$ : Let's first evaluate $g(x)$ at $x = 1$ and $x = 4$ to determine the values of $g(1)$ and $g(4)$ .
$g(1) = 5 - e^{1} = 5 - e$
$g(4) = 5 - e^{4}$
Check Straddling of $0$ : Now we need to check if $g(1)$ and $g(4)$ straddle $0$ , which means one should be positive and the other should be negative for the Intermediate Value Theorem to guarantee a solution for $g(x) = 0$ between $x = 1$ and $x = 4$ . Since $e$ is approximately $2.71828$ , we can see that $g(1) = 5 - e$ is positive because $5$ is greater than $e$ .
Evaluate $g(4)$ : Next, we need to evaluate whether g(4) < 0. $g(4) = 5 - e^{4}$ , and since $e^{4}$ is much larger than $5$ , $g(4)$ will be negative.
Apply Intermediate Value Theorem: Since $g(1)$ is positive and $g(4)$ is negative, and $g(x)$ is continuous (as it is a combination of continuous functions: a constant, subtraction, and an exponential function), the Intermediate Value Theorem applies.
Incomplete Justification: Sean's justification is incomplete because he did not explicitly establish that $g(1)$ is greater than $0$ and $g(4)$ is less than $0$ , which is necessary to apply the Intermediate Value Theorem to ensure that a zero of $g(x)$ exists between $x = 1$ and $x = 4$ .