You’ve uploaded a screenshot from a SMART Notebook lesson titled
“3.3 Perform Function Operations and Composition”, specifically
Example 1, which asks to:
> Let \( f(x) = 4x^{1/2} \) and \( g(x) = -9x^{1/2} \). Find the following:
> a. \( f(x) + g(x) \)
> b. \( f(x) - g(x) \)
> c. the domains of \( f + g \) and \( f - g \)
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✔ Let’s solve each part step by step and explain the reasoning.
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## 🔹 Part (a): \( f(x) + g(x) \)
We are given:
- \( f(x) = 4x^{1/2} \)
- \( g(x) = -9x^{1/2} \)
So,
\[
f(x) + g(x) = 4x^{1/2} + (-9x^{1/2}) = (4 - 9)x^{1/2} = -5x^{1/2}
\]
✔ Answer: \( \boxed{-5x^{1/2}} \)
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## 🔹 Part (b): \( f(x) - g(x) \)
\[
f(x) - g(x) = 4x^{1/2} - (-9x^{1/2}) = 4x^{1/2} + 9x^{1/2} = (4 + 9)x^{1/2} = 13x^{1/2}
\]
✔ Answer: \( \boxed{13x^{1/2}} \)
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## 🔹 Part (c): Domain of \( f + g \) and \( f - g \)
📌 What is the domain?
The
domain is the set of all real numbers \( x \) for which the function is defined.
Both \( f(x) \) and \( g(x) \) involve \( x^{1/2} \), which is the same as \( \sqrt{x} \).
👉 The square root function \( \sqrt{x} \) is
only defined for \( x \geq 0 \) in the real number system.
Since both \( f \) and \( g \) are defined only when \( x \geq 0 \), their sum and difference will also be defined only where
both functions are defined — that is, their
common domain.
Thus, the domain of \( f + g \) and \( f - g \) is:
\[
\boxed{[0, \infty)}
\]
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##
✔ Final Answers:
a. \( f(x) + g(x) = \boxed{-5x^{1/2}} \)
b. \( f(x) - g(x) = \boxed{13x^{1/2}} \)
c. Domain of both \( f + g \) and \( f - g \): \( \boxed{[0, \infty)} \)
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## 💡 Key Concept:
When adding or subtracting functions, the
domain of the resulting function is the
intersection of the domains of the original functions — meaning, it includes only the values of \( x \) where
both functions are defined.
In this case, since both functions have the same domain \( [0, \infty) \), the domain of their sum and difference is also \( [0, \infty) \).
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✔️ Your handwritten work in the image is
correct! Great job!
Parent Tip: Review the logic above to help your child master the concept of function operations and composition worksheet.
