# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-7/14

Exercises: Try Out - Set 7: Real exam simulation to sharpen your skills and build confidence. - Problem 14

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## Exercise 14

### Question

import { Graph } from "../graph";

export const metadata = {
  title: "Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Given <InlineMath math="f(x) = 5 - x" /> and <InlineMath math="g(x) = 2^x - 1" />. Determine which of the following statements are correct:

1. <InlineMath math="g(x)" /> is a linear line with a gradient of <InlineMath math="2" />.
2. <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="x > 0" />.
3. <InlineMath math="f(x)" /> is above <InlineMath math="g(x)" /> for all values of <InlineMath math="x" />.
4. The graphs of <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="(2, 3)" />.

<Graph title="Function Graph" description={<>Graph of functions <InlineMath math="f(x)" /> and <InlineMath math="g(x)" />.</>} />


### Choices

- [ ] $$(1), (2), \text{ and } (3) \text{ are correct}.$$
- [ ] $$(1) \text{ and } (3) \text{ are correct}.$$
- [x] $$(2) \text{ and } (4) \text{ are correct}.$$
- [ ] $$\text{Only } (4) \text{ is correct}.$$
- [ ] $$\text{All are correct}.$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 14",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

We will analyze each statement one by one based on the functions <InlineMath math="f(x) = 5 - x" /> and <InlineMath math="g(x) = 2^x - 1" />.

#### Analysis of Statement 1

Statement: <InlineMath math="g(x)" /> is a linear line with a gradient of <InlineMath math="2" />.

The function <InlineMath math="g(x) = 2^x - 1" /> is an exponential function, not a linear function. A linear function has the general form <InlineMath math="y = mx + c" />, whereas this is an exponential form. Therefore, statement <InlineMath math="(1)" /> is **incorrect**.

#### Analysis of Statement 2

Statement: <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="x > 0" />.

To find the intersection point, we equate the two functions:

<MathContainer>
  <BlockMath math="f(x) = g(x)" />
  <BlockMath math="5 - x = 2^x - 1" />
</MathContainer>

If we substitute <InlineMath math="x = 2" />:

<MathContainer>
  <BlockMath math="5 - 2 = 3" />
  <BlockMath math="2^2 - 1 = 4 - 1 = 3" />
</MathContainer>

Both functions have the same value of <InlineMath math="3" /> when <InlineMath math="x = 2" />. Since <InlineMath math="2 > 0" />, statement <InlineMath math="(2)" /> is **correct**.

#### Analysis of Statement 3

Statement: <InlineMath math="f(x)" /> is above <InlineMath math="g(x)" /> for all values of <InlineMath math="x" />.

Let's check a value <InlineMath math="x > 2" />, for example <InlineMath math="x = 3" />:

<MathContainer>
  <BlockMath math="f(3) = 5 - 3 = 2" />
  <BlockMath math="g(3) = 2^3 - 1 = 8 - 1 = 7" />
</MathContainer>

Here it is seen that <InlineMath math="g(3) > f(3)" />, which means the curve <InlineMath math="g(x)" /> is above <InlineMath math="f(x)" />. Therefore, the statement that <InlineMath math="f(x)" /> is always above <InlineMath math="g(x)" /> is incorrect. Statement <InlineMath math="(3)" /> is **incorrect**.

#### Analysis of Statement 4

Statement: The graphs of <InlineMath math="f(x)" /> and <InlineMath math="g(x)" /> intersect at <InlineMath math="(2, 3)" />.

As proven in the analysis of statement <InlineMath math="(2)" />, both graphs intersect when <InlineMath math="x = 2" /> and yield the value <InlineMath math="y = 3" />. Thus, the intersection point is <InlineMath math="(2, 3)" />. Statement <InlineMath math="(4)" /> is **correct**.

#### Conclusion

The correct statements are <InlineMath math="(2)" /> and <InlineMath math="(4)" />.


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