# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-6/20

Exercises: Try Out - Set 6: Real exam simulation to sharpen your skills and build confidence. - Problem 20

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## Exercise 20

### Question

export const metadata = {
  title: "Problem 20",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

If <InlineMath math="f(x) = 5 - 4x^2" /> and <InlineMath math="g(x) = \frac{3x - 5}{2}" />, then <InlineMath math="(f \circ g)(x) =" /> ...


### Choices

- [ ] $$-9x^2 + 30x + 30$$
- [x] $$-9x^2 + 30x - 20$$
- [ ] $$-9x^2 - 30x - 20$$
- [ ] $$-9x^2 + 30$$
- [ ] $$-9x^2 - 20$$

### Answer & Explanation

export const metadata = {
  title: "Problem 20 Solution",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the functions:

<MathContainer>
  <BlockMath math="f(x) = 5 - 4x^2" />
  <BlockMath math="g(x) = \frac{3x - 5}{2}" />
</MathContainer>

We are asked to find the composition function <InlineMath math="(f \circ g)(x)" />.

By definition, <InlineMath math="(f \circ g)(x) = f(g(x))" />. This means we substitute the entire function <InlineMath math="g(x)" /> into the variable <InlineMath math="x" /> in function <InlineMath math="f(x)" />.

<MathContainer>
  <BlockMath math="\begin{aligned} (f \circ g)(x) &= f\left(\frac{3x - 5}{2}\right) \\ &= 5 - 4\left(\frac{3x - 5}{2}\right)^2 \end{aligned}" />
</MathContainer>

Next, expand the squared fraction:

<MathContainer>
  <BlockMath math="\begin{aligned} &= 5 - 4\left(\frac{(3x - 5)^2}{2^2}\right) \\ &= 5 - 4\left(\frac{9x^2 - 30x + 25}{4}\right) \end{aligned}" />
</MathContainer>

Notice that the <InlineMath math="4" /> in the numerator and denominator cancel out:

<MathContainer>
  <BlockMath math="\begin{aligned} &= 5 - (9x^2 - 30x + 25) \end{aligned}" />
</MathContainer>

Distribute the negative sign:

<MathContainer>
  <BlockMath math="\begin{aligned} &= 5 - 9x^2 + 30x - 25 \\ &= -9x^2 + 30x - 20 \end{aligned}" />
</MathContainer>

Thus, <InlineMath math="(f \circ g)(x) = -9x^2 + 30x - 20" />.


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