1.1 Q-1

Questions Statement

For: (a) $f(x)=x^{2}-x$ (b) $f(x)=\sqrt{x+4}$.

Find:

1. $f(-2)$,
2. $f(0)$,
3. $f(x-1)$,
4. $f\left(x^{2}+4\right)$.

Solution

(a) For $f(x) = x^2 - x$:

1. Finding $f(-2)$:
   • We are asked to evaluate $f(x)$ at $x = -2$. To do this, we substitute $-2$ into the function $f(x) = x^2 - x$:

$$f(-2) = (-2)^2 - (-2) = 4 + 2 = 6$$

• Therefore, $f(-2) = 6$.

2. Finding $f(0)$:
   • Next, we evaluate $f(x)$ at $x = 0$. Substituting $0$ into the function:

$$f(0) = (0)^2 - (0) = 0 - 0 = 0$$

• So, $f(0) = 0$.

3. Finding $f(x-1)$:
   • Now, we need to evaluate $f(x-1)$, which means substituting $(x - 1)$ in place of $x$ in the function $f(x) = x^2 - x$. This gives:

$$f(x-1) = \left[(x-1)^2 - (x-1)\right]$$

• First, expand $(x-1)^2$:

$$(x-1)^2 = x^2 - 2x + 1$$

• So, the expression for $f(x-1)$ becomes:

$$f(x-1) = x^2 - 2x + 1 - x + 1 = x^2 - 3x + 2$$

• Thus, $f(x-1) = x^2 - 3x + 2$.

4. Finding $f(x^2 + 4)$:
   • Next, we evaluate $f(x^2 + 4)$ by substituting $(x^2 + 4)$ for $x$ in the function $f(x) = x^2 - x$:

$$f(x^2 + 4) = \left[(x^2 + 4)^2 - (x^2 + 4)\right]$$

• First, expand $(x^2 + 4)^2$:

$$(x^2 + 4)^2 = x^4 + 8x^2 + 16$$

• Now substitute and simplify:

$$f(x^2 + 4) = x^4 + 8x^2 + 16 - x^2 - 4 = x^4 + 7x^2 + 12$$

• Therefore, $f(x^2 + 4) = x^4 + 7x^2 + 12$.

(b) For $f(x) = \sqrt{x+4}$:

1. Finding $f(-2)$:
   • We substitute $-2$ into the function $f(x) = \sqrt{x + 4}$:

$$f(-2) = \sqrt{-2 + 4} = \sqrt{2}$$

• So, $f(-2) = \sqrt{2}$.

2. Finding $f(0)$:
   • Now, we evaluate $f(x)$ at $x = 0$:

$$f(0) = \sqrt{0 + 4} = \sqrt{4} = 2$$

• Therefore, $f(0) = 2$.

3. Finding $f(x-1)$:
   • For $f(x-1)$, substitute $(x - 1)$ in place of $x$ in the function $f(x) = \sqrt{x + 4}$:

$$f(x-1) = \sqrt{(x-1) + 4} = \sqrt{x + 3}$$

• So, $f(x-1) = \sqrt{x + 3}$.

4. Finding $f(x^2 + 4)$:
   • For $f(x^2 + 4)$, substitute $(x^2 + 4)$ in place of $x$:

$$f(x^2 + 4) = \sqrt{x^2 + 4 + 4} = \sqrt{x^2 + 8}$$

• Therefore, $f(x^2 + 4) = \sqrt{x^2 + 8}$.

Key Formulas or Methods Used

• Function Evaluation: Substituting values or expressions into a function to find the result.
• Algebraic Manipulation: Expanding and simplifying expressions, such as $(x - 1)^2$ and $(x^2 + 4)^2$.
• Square Root Function: Applying the square root formula for the given function $f(x) = \sqrt{x+4}$.

Summary of Steps

1. For part (a), evaluate the function $f(x) = x^2 - x$ at the given values and expressions.
2. For part (b), evaluate the function $f(x) = \sqrt{x+4}$ at the specified values.
3. Simplify the resulting expressions for $f(x-1)$, $f(x^2+4)$, and other terms as needed.