Powers Grant
07/06/2023 · High School
9) Find the domain of \( g(x)=\sqrt{2 x^{2}+7 x-4} \) \[ \begin{array}{l} x^{2}+7 x-8 \\ (x-1 / 2(x+8)\end{array} \]
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To find the domain of the function \( g(x) = \sqrt{2x^2 + 7x - 4} \), we need to determine the values of \( x \) for which the expression inside the square root is non-negative, as the square root of a negative number is not a real number.
The expression inside the square root is \( 2x^2 + 7x - 4 \). We need to find the values of \( x \) for which this expression is greater than or equal to 0.
Let's solve the inequality \( 2x^2 + 7x - 4 \geq 0 \) to find the domain of the function.
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(2x^{2}+7x-4\geq 0\)
- step1: Rewrite the expression:
\(2x^{2}+7x-4=0\)
- step2: Factor the expression:
\(\left(x+4\right)\left(2x-1\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&x+4=0\\&2x-1=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=-4\\&x=\frac{1}{2}\end{align}\)
- step5: Determine the test intervals:
\(\begin{align}&x<-4\\&-4\frac{1}{2}\end{align}\)
- step6: Choose a value:
\(\begin{align}&x_{1}=-5\\&x_{2}=-2\\&x_{3}=2\end{align}\)
- step7: Test the chosen value:
\(\begin{align}&x<-4\textrm{ }\textrm{is the solution}\\&-4\frac{1}{2}\textrm{ }\textrm{is the solution}\end{align}\)
- step8: Include the critical value:
\(\begin{align}&x\leq -4\textrm{ }\textrm{is the solution}\\&x\geq \frac{1}{2}\textrm{ }\textrm{is the solution}\end{align}\)
- step9: The final solution is \(x \in \left(-\infty,-4\right]\cup \left[\frac{1}{2},+\infty\right):\)
\(x \in \left(-\infty,-4\right]\cup \left[\frac{1}{2},+\infty\right)\)
The solution to the inequality \(2x^2 + 7x - 4 \geq 0\) is \(x \in (-\infty, -4] \cup [\frac{1}{2}, +\infty)\).
Therefore, the domain of the function \(g(x) = \sqrt{2x^2 + 7x - 4}\) is \(x \in (-\infty, -4] \cup [\frac{1}{2}, +\infty)\).
Quick Answer
The domain of \(g(x) = \sqrt{2x^2 + 7x - 4}\) is \(x \in (-\infty, -4] \cup [\frac{1}{2}, +\infty)\).
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit