QUESTION 3 Suppose the functions $$ f, g, h $$ and $$ l $$ are defined as follows: $$ \begin{array}{l} f(x)=4 x^{2}-5 x+1 \ g(x)=2 \sqrt{2-\frac{x}{2}}-x \ h(x)=-\frac{1}{2} x+3 \ l(x)=\log _{4}(x+3)-\log _{4}(x-2) .\end{array} $$ (a) Write down $$ D_{f} $$ and solve the inequality $$ f(x) \leq 0 $$. (b) Solve the equation $$ g(x)=-4 $$. (c) Solve the equation $$ 4^{h(x)}=8 $$. (d) Write down $$ D_{l} $$ and solve the inequality $$ l(x)=\frac{1}{2} $$. [18] (4)

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Let's solve **Question 3 (a)** step by step.

### **(a) Find the Domain $$ D_f $$ and Solve the Inequality $$ f(x) \leq 0 $$**

**1. Determine the Domain $$ D_f $$:**

The function $$ f(x) = 4x^2 - 5x + 1 $$ is a quadratic polynomial. Polynomial functions are defined for all real numbers. Therefore,

$$
D_f = \mathbb{R} \quad \text{(All real numbers)}
$$

**2. Solve the Inequality $$ f(x) \leq 0 $$:**

We need to find all real numbers $$ x $$ such that:

$$
4x^2 - 5x + 1 \leq 0
$$

**a. Find the Roots of the Equation $$ 4x^2 - 5x + 1 = 0 $$:**

Using the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, where $$ a = 4 $$, $$ b = -5 $$, and $$ c = 1 $$:

$$
x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} = \frac{5 \pm \sqrt{25 - 16}}{8} = \frac{5 \pm 3}{8}
$$

So, the roots are:

$$
x_1 = \frac{5 + 3}{8} = \frac{8}{8} = 1
$$
$$
x_2 = \frac{5 - 3}{8} = \frac{2}{8} = \frac{1}{4}
$$

**b. Analyze the Sign of $$ f(x) $$:**

Since the coefficient of $$ x^2 $$ is positive ($$ 4 > 0 $$), the parabola opens upwards. Therefore, $$ f(x) \leq 0 $$ between the roots.

**c. Interval of Solution:**

$$
\frac{1}{4} \leq x \leq 1
$$

**3. Final Answer:**

- **Domain $$ D_f $$:**

$$
D_f = \mathbb{R} \quad \text{(All real numbers)}
$$

- **Solution to $$ f(x) \leq 0 $$:**

$$
x \in \left[ \frac{1}{4}, \ 1 \right]
$$
**Domain $$ D_f $$: All real numbers.** **Solution to $$ f(x) \leq 0 $$:** $$ \frac{1}{4} \leq x \leq 1 $$

QUESTION 3
Suppose the functions and are defined as follows:

(a) Write down and solve the inequality .
(b) Solve the equation .
© Solve the equation .
(d) Write down and solve the inequality .
[18] (4)

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QUESTION 3 Suppose the functions and are defined as follows: (a) Write down and solve the inequality . (b) Solve the equation . © Solve the equation . (d) Write down and solve the inequality . [18] (4)