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Q5(ii):
Find the range of each of the following functions. (ii) $f(x) = x^2 + 2$, $x$ is a real number.
Solution :
Step 1: Identifying the Domain and Function Definition
Given the function $f(x) = x^2 + 2$, where $x \in \mathbb{R}$ (the set of all real numbers). The domain is defined as the set of all possible input values for $x$, which is $(-\infty, \infty)$.
Step 2: Analyzing the Square Property of Real Numbers
By the definition of real numbers, the square of any real number is always non-negative [Axiom of non-negativity in ordered fields]:
For all $x \in \mathbb{R}$, $x^2 \geq 0$.
Step 3: Transforming the Inequality to Match the Function
To determine the range, we must transform $x^2$ into $x^2 + 2$ by applying the Addition Property of Inequalities [Property of Ordered Fields]:
Add $2$ to both sides of the inequality $x^2 \geq 0$:
$x^2 + 2 \geq 0 + 2$
$x^2 + 2 \geq 2$
Step 4: Establishing the Range
Since $f(x) = x^2 + 2$, we conclude that $f(x) \geq 2$. The range is the set of all output values $y$ such that $y = f(x)$. Therefore, the range of the function is the interval $[2, \infty)$.
Final Solution: The range of the function $f(x) = x^2 + 2$ is $[2, \infty)$.
More Questions from Class 11 Mathematics Relations and Functions EXERCISE 2.3
- Q1(i): Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. (i) \{(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)\}
- Q1(ii): Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. (ii) \{(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)\}
- Q1(iii): Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. (iii) \{(1,3), (1,5), (2,5)\}
- Q2(i): Find the domain and range of the following real functions: (i) $f(x) = –|x|$
- Q2(ii): Find the domain and range of the following real functions: (ii) $f(x) = \sqrt{9-x^2}$.
- Q3(i): A function $f$ is defined by $f(x) = 2x –5$. Write down the values of (i) $f(0)$
- Q3(ii): A function $f$ is defined by $f(x) = 2x –5$. Write down the values of (ii) $f(7)$
- Q3(iii): A function $f$ is defined by $f(x) = 2x –5$. Write down the values of (iii) $f(–3)$
- Q4(i): The function ‘$t$’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(C) = \frac{9C}{5} + 32$. Find (i) $t(0)$
- Q4(ii): The function ‘$t$’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(C) = \frac{9C}{5} + 32$. Find (ii) $t(28)$
- Q4(iii): The function ‘$t$’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(C) = \frac{9C}{5} + 32$. Find (iii) $t(–10)$
- Q4(iv): The function ‘$t$’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(C) = \frac{9C}{5} + 32$. Find (iv) The value of $C$, when $t(C) = 212$.
- Q5(i): Find the range of each of the following functions. (i) $f(x) = 2 – 3x, x \in \mathbb{R}, x > 0$.
- Q5(iii): Find the range of each of the following functions. (iii) $f(x) = x$, $x$ is a real number.
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