Sure, let's break down the function [tex]\( f(x) = \sqrt{49 - x^2} + \sqrt{x^2 - 1} \)[/tex] into detailed steps for better understanding.
1. Identify Individual Components:
- The first part of the function is [tex]\( \sqrt{49 - x^2} \)[/tex].
- The second part of the function is [tex]\( \sqrt{x^2 - 1} \)[/tex].
2. Consider the Domain of the Function:
- The expression [tex]\( \sqrt{49 - x^2} \)[/tex] must have a non-negative argument inside the square root. Hence, [tex]\( 49 - x^2 \geq 0 \)[/tex]. Solving [tex]\( 49 - x^2 \geq 0 \)[/tex], we get [tex]\( 49 \geq x^2 \)[/tex] or [tex]\( -7 \leq x \leq 7 \)[/tex].
- Similarly, the expression [tex]\( \sqrt{x^2 - 1} \)[/tex] must also have a non-negative argument. Thus, [tex]\( x^2 - 1 \geq 0 \)[/tex]. Solving [tex]\( x^2 - 1 \geq 0 \)[/tex], we get [tex]\( x^2 \geq 1 \)[/tex], meaning [tex]\( x \leq -1 \)[/tex] or [tex]\( x \geq 1 \)[/tex].
3. Determine the Common Domain:
- Combining these two conditions, the valid values for [tex]\( x \)[/tex] that satisfy both [tex]\( \sqrt{49 - x^2} \geq 0 \)[/tex] and [tex]\( \sqrt{x^2 - 1} \geq 0 \)[/tex] are [tex]\( -7 \leq x \leq -1 \)[/tex] or [tex]\( 1 \leq x \leq 7 \)[/tex].
4. Expression Breakdown:
- For values of [tex]\( x \)[/tex] within the domain [tex]\( -7 \leq x \leq -1 \)[/tex] and [tex]\( 1 \leq x \leq 7 \)[/tex], the function [tex]\( f(x) \)[/tex] is defined and given by:
[tex]\[
f(x) = \sqrt{49 - x^2} + \sqrt{x^2 - 1}
\][/tex]
5. Graphical Representation and Understanding:
- The function [tex]\( f(x) \)[/tex] can be graphed to understand its behavior. Given the constraints on [tex]\( x \)[/tex], [tex]\( f(x) \)[/tex] will be a combination of two symmetrical, non-linear segments resulting from the sum of the two square root terms. It's possible to visualize this function by plotting it within the given domain.
In summary, the function [tex]\( f(x) = \sqrt{49 - x^2} + \sqrt{x^2 - 1} \)[/tex] is defined for [tex]\( x \in [-7, -1] \cup [1, 7] \)[/tex]. These constraints are necessary to ensure that both square root terms are real and non-negative.
