Certainly! Let's go through the process of finding the value of the function [tex]\( f(x) = \sqrt{4x^2 + 6x - 3} \)[/tex] at a specific value of [tex]\( x \)[/tex]. Given the result, we'll illustrate how the numerical solution was achieved.
We need to calculate [tex]\( f(1) \)[/tex], which involves evaluating the function at [tex]\( x = 1 \)[/tex].
1. Substitute [tex]\( x = 1 \)[/tex] into the function:
[tex]\[
f(1) = \sqrt{4(1)^2 + 6(1) - 3}
\][/tex]
2. Simplify the expression inside the square root:
[tex]\[
4(1)^2 = 4 \quad \text{(since \( 1^2 = 1 \))}
\][/tex]
[tex]\[
6(1) = 6
\][/tex]
[tex]\[
4 + 6 - 3 = 7
\][/tex]
3. Take the square root of the simplified result:
[tex]\[
f(1) = \sqrt{7}
\][/tex]
4. State the result:
After evaluating [tex]\( \sqrt{7} \)[/tex] using a calculator or through precision arithmetic, we get:
[tex]\[
f(1) \approx 2.6457513110645907
\][/tex]
Therefore, the value of the function [tex]\( f(x) = \sqrt{4x^2 + 6x - 3} \)[/tex] at [tex]\( x = 1 \)[/tex] is approximately [tex]\( 2.6457513110645907 \)[/tex].
