Question
Suppose that the functions \( f \) and \( g \) are defined for all real numbers \( x \) as follows. \[ \begin{array}{l}f(x)=3 x \\ g(x)=4 x^{2} \\ \text { Write the expressions for }(g-f)(x) \text { and }(g \cdot f)(x) \text { and evaluate }(g+f)(2) \text {. } \\ \begin{array}{l}(g-f)(x)=\square \\ (g \cdot f)(x)=\square\end{array}\end{array} . \]
Ask by Romero Tyler. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
(g - f)(x) = 4x^{2} - 3x
\]
\[
(g \cdot f)(x) = 12x^{3}
\]
\[
(g + f)(2) = 22
\]
Solution
Let's evaluate the expressions step by step.
Given:
\[
f(x) = 3x \quad \text{and} \quad g(x) = 4x^{2}
\]
1. **Finding \((g - f)(x)\):**
\[
(g - f)(x) = g(x) - f(x) = 4x^{2} - 3x
\]
2. **Finding \((g \cdot f)(x)\):**
\[
(g \cdot f)(x) = g(x) \times f(x) = (4x^{2}) \times (3x) = 12x^{3}
\]
3. **Evaluating \((g + f)(2)\):**
\[
(g + f)(x) = g(x) + f(x) = 4x^{2} + 3x
\]
\[
(g + f)(2) = 4(2)^{2} + 3(2) = 4 \times 4 + 6 = 16 + 6 = 22
\]
**Summary of Results:**
\[
\begin{array}{l}
(g - f)(x) = 4x^{2} - 3x \\
(g \cdot f)(x) = 12x^{3}
\end{array}
\]
\[
(g + f)(2) = 22
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the expressions for \((g - f)(x)\) and \((g \cdot f)(x)\), we can start by defining these operations using the functions given. 1. **Finding \((g - f)(x)\):** \[ (g - f)(x) = g(x) - f(x) = 4x^2 - 3x \] 2. **Finding \((g \cdot f)(x)\):** \[ (g \cdot f)(x) = g(x) \cdot f(x) = (4x^2)(3x) = 12x^3 \] Now, let’s evaluate \((g + f)(2)\): \[ g(2) = 4(2^2) = 4 \cdot 4 = 16, \quad f(2) = 3(2) = 6 \] So, \[ (g + f)(2) = g(2) + f(2) = 16 + 6 = 22 \] In summary, we have: \[ (g-f)(x) = 4x^2 - 3x, \quad (g \cdot f)(x) = 12x^3, \quad (g + f)(2) = 22 \]
