To find \((f + \frac{g}{h})(x)\), we need to first compute \(\frac{g(x)}{h(x)}\) and then add \(f(x)\) to that result.
Given:
- \(f(x) = x^4 + 7x^2 - 9\)
- \(g(x) = 8x^2 + 4x\)
- \(h(x) = 2x\)
Step 1: Compute \(\frac{g(x)}{h(x)}\)
\[ \frac{g(x)}{h(x)} = \frac{8x^2 + 4x}{2x} \]
We can simplify this expression:
\[ \frac{g(x)}{h(x)} = \frac{8x^2}{2x} + \frac{4x}{2x} = 4x + 2 \]
Step 2: Calculate \((f + \frac{g}{h})(x)\)
Now, we add \(f(x)\) and \(\frac{g(x)}{h(x)}\):
\[ (f + \frac{g}{h})(x) = f(x) + (4x + 2) \]
Substituting for \(f(x)\):
\[ (f + \frac{g}{h})(x) = \left(x^4 + 7x^2 - 9\right) + (4x + 2) \]
Combining like terms:
\[ (f + \frac{g}{h})(x) = x^4 + 7x^2 + 4x - 7 \]
Step 3: Determine the domain of \((f + \frac{g}{h})(x)\)
The domain of this expression is determined by the function \(\frac{g(x)}{h(x)}\), specifically by \(h(x)\), since we cannot divide by zero:
\[ h(x) = 2x \Rightarrow h(x) \neq 0 \Rightarrow x \neq 0 \]
Thus, the domain of \((f + \frac{g}{h})(x)\) is:
\[ \text{Domain: } x \neq 0 \]
Final result:
\((f + \frac{g}{h})(x) = x^4 + 7x^2 + 4x - 7\) and the domain is \(x \neq 0\).
Based on the provided options in your question, the correct response would be:
\((f+g÷h)(x)=x^4+7x^2+4x−7\) and the domain of \((f+g÷h)(x)\) is \(x \neq 0\).
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