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