Functions (Range) – Problems

Functions (Range) – Problems

1. The range of the function $$f(x)=\frac{x+2}{|x+2|}$$ =?

Solution: $$f(x)=\frac{x+2}{|x+2|}$$,

f(x)=\left\{ \begin{align} & -1,\,\,\,x<-2 \\ & 1,\,\,\,\,\,x>-2 \\ \end{align} \right.,

∴ Range of f (x) is {-1, 1}.

2. If x is real then value of the expression $$\frac{{{x}^{2}}+14x+9}{{{x}^{2}}+2x+3}$$ lies between ?

Solution: $$\frac{{{x}^{2}}+14x+9}{{{x}^{2}}+2x+3}$$,

Let us y = $$\frac{{{x}^{2}}+14x+9}{{{x}^{2}}+2x+3}$$.

⇒ x² + 14x + 9 = yx² + 2xy + 3y

⇒ x² (y – 1) + 2x (y – 7) + (3y – 9) = 0

Since x is real

∴ 4 (y – 7)² – 4 (3y – 9) (y – 1) > 0 (since b² – 4ac > 0)

⇒ 4 (y² + 49 – 14y) – 4 (3y² + 9 – 12 y) > 0

⇒ 4y² + 196 – 56 y – 12y² – 36 + 48y > 0

⇒ 8y² + 8y – 160 < 0

y² + y – 20 < 0

(y + 5) (y – 4) < 0

Given expression lies between -5, 4

3. The range of function the range of function f (x) = x² – 6x + 7 =?

Solution: x² – 6x + 7 = (x – 3)² – 2

Minimum value is -2 and maximum ∞

Hence range of function is [-2, ∞)

4. The inverse of the function $$\frac{{{10}^{x}}-{{10}^{-x}}}{{{10}^{x}}+{{10}^{-x}}}$$ =?

Solution: $$y=\frac{{{10}^{x}}-{{10}^{-x}}}{{{10}^{x}}+{{10}^{-x}}}$$ ,

⇒ $$x=\frac{1}{2}{{\log }_{10}}\left( \frac{1+y}{1-y} \right)$$,

Let y = f (x) ⇒ x = f⁻¹ (y)

⇒ $${{f}^{-1}}(y)=\frac{1}{2}{{\log }_{10}}\left( \frac{1+y}{1-y} \right)$$,

⇒ $${{f}^{-1}}(x)=\frac{1}{2}{{\log }_{10}}\left( \frac{1+x}{1-x} \right)$$.