Question

let $f(x) = 2^{5 - 2x}$. below is ibrahims attempt to write a formal justification for the fact that the equation $f(x) = 10$ has a solution where $-1 \leq x \leq 4$. is ibrahims justification complete? if not, why? ibrahims justification: $f$ is defined for all real numbers, and exponential functions are continuous at all points in their domains. also, $f(-1) = 128$ and $f(4) = \frac{1}{8}$, so 10 is between $f(-1)$ and $f(4)$. so, according to the intermediate value theorem, $f(x) = 10$ must have a solution at some point between $x = -1$ and $x = 4$. choose 1 answer: a yes, ibrahims justification is complete. b no, ibrahim didnt establish that 10 is between $f(-1)$ and $f(4)$. c no, ibrahim didnt establish that $f$ is continuous.

Explanation:

To use the Intermediate Value Theorem (IVT), we need two things: the function is continuous on the interval \([a, b]\) and the value \(k\) (here \(k = 10\)) is between \(f(a)\) and \(f(b)\) (i.e., either \(f(a)\leq k\leq f(b)\) or \(f(b)\leq k\leq f(a)\)).

• Ibrahim stated \(f(- 1)=128\) and \(f(4)=\frac{1}{8}\). But he just said "10 is between \(f(-1)\) and \(f(4)\)" without checking the order. Since \(f(4)=\frac{1}{8}<10 < 128=f(-1)\), he needs to establish that \(f(4)\leq10\leq f(-1)\) (or show the inequality direction) to properly apply IVT. Option A is wrong because the justification is not complete. Option C is wrong because he did establish continuity (exponential functions are continuous on their domain, and he said \(f\) is defined for all real numbers, so domain is \(\mathbb{R}\), hence continuous on \([-1,4]\)). So the error is that he didn't properly establish that 10 is between \(f(-1)\) and \(f(4)\) by checking the inequality direction (i.e., that \(f(4)\leq10\leq f(-1)\)).

Answer:

B. No, Ibrahim didn't establish that 10 is between \(f(-1)\) and \(f(4)\).