Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Let h(x)=(1)/(4)x^(3)+2x-1 and let g be the inverse function of h. Notice that h(2)=5. g^(')(5)=

Let h(x)=14x3+2x−1 h(x)=\frac{1}{4} x^{3}+2 x-1 and let g g be the inverse function of h h . Notice that h(2)=5 h(2)=5 .\newlineg′(5)= g^{\prime}(5)=

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Transformations of quadratic functionsright chevron icon

Full solution

Q. Let h(x)=14x3+2x−1 h(x)=\frac{1}{4} x^{3}+2 x-1 and let g g be the inverse function of h h . Notice that h(2)=5 h(2)=5 .\newlineg′(5)= g^{\prime}(5)=
  1. Use Inverse Function Formula: To find g′(5)g'(5), we need to use the formula for the derivative of the inverse function: g′(h(x))=1h′(g(x)).g'(h(x)) = \frac{1}{h'(g(x))}.
  2. Find h′(x)h'(x): First, we need to find h′(x)h'(x), which is the derivative of h(x)h(x) with respect to xx.\newlineh′(x)=14⋅3x2+2h'(x) = \frac{1}{4} \cdot 3x^2 + 2.
  3. Calculate h′(2)h'(2): Now we plug in x=2x = 2 into h′(x)h'(x) to find h′(2)h'(2).\newlineh′(2)=(14)⋅3⋅22+2=(14)⋅3⋅4+2=3+2=5h'(2) = (\frac{1}{4}) \cdot 3 \cdot 2^2 + 2 = (\frac{1}{4}) \cdot 3 \cdot 4 + 2 = 3 + 2 = 5.
  4. Apply Inverse Function Formula: Now we use the formula for the derivative of the inverse function. g′(5)=1h′(2)=15g'(5) = \frac{1}{h'(2)} = \frac{1}{5}.

More problems from Transformations of quadratic functions

Question
The function h h is defined over the real numbers. This table gives a few values of h h .\newline\begin{tabular}{lllll}\newlinex x & −6-6.11 & −6-6.0101 & −6-6.001001 & −5-5.99 \\\newline\hlineh(x) h(x) & −0-0.2525 & −0-0.7474 & −0-0.9898 & −1-1.00\newline\end{tabular}\newlineWhat is a reasonable estimate for lim⁡x→−6h(x) \lim _{x \rightarrow-6} h(x) ?\newlineChoose 11 answer:\newline(A) −6-6\newline(B) −2-2\newline(C) −1-1\newline(D) The limit doesn't exist
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the amplitude of\newlineg(x)=−2sin⁡(π2x−3)+5? g(x)=-2 \sin \left(\frac{\pi}{2} x-3\right)+5 ? \newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the amplitude of y=5sin⁡(4x−2)−3 y=5 \sin (4 x-2)-3 ?\newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the amplitude of y=−3cos⁡(πx+2)−6? y=-3 \cos (\pi x+2)-6 ? \newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the amplitude of y=3sin⁡(2x−1)+4 y=3 \sin (2 x-1)+4 ?\newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the period of the function h(x)=−3cos⁡(πx+2)−6? h(x)=-3 \cos (\pi x+2)-6 ? \newlineGive an exact value.\newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the period of the function\newlineh(x)=5sin⁡(4x−2)−3 ?  h(x)=5 \sin (4 x-2)-3 \text { ? } \newlineGive an exact value.\newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the period of the function g(x)=2cos⁡(7x+5)+1? g(x)=2 \cos (7 x+5)+1 ? \newlineGive an exact value.\newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the period of the function f(x)=3sin⁡(2x−1)+4 f(x)=3 \sin (2 x-1)+4 ?\newlineGive an exact value.\newlineunits
Get tutor helpright-arrow

Posted 10 months ago

Question
What is the period of the function f(x)=−4cos⁡(5x−9)−7 f(x)=-4 \cos (5 x-9)-7 ?\newlineGive an exact value.\newlineunits
Get tutor helpright-arrow

Posted 10 months ago

View all (14)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant