Simplifying Composite Functions
Read this topicExplanations 4
- (Video) Composition of Functions by PatrickJMT Here's a video by PatrickJMT showing you what a composition of a function is. Here is a pair of practice questiosn to accompany th…
By Alex Federspiel, Joshua Siktar, Anusha Rahman
- You can use your substitution abilities to simplify a composition of functions! When we're simplifying f(g(x)),f(g(x)),f(g(x)), we substitute our g(x)g(x)g(x) function into our …
By Caroline K, Alex Federspiel, Joshua Siktar, Anusha Rahman
- The concept of a composite function is that it helps answer the question, "what happens when you plug a function into a function?" Well, the answer is that you get a different f…
By Joshua Siktar, Anusha Rahman
- Simplifying Composite Functions When we simplify composite functions, we are working with two functions where one function is inside the other function. We've covered this previ…
By Anusha Rahman
Problems 58
- For real numbers xxx and yyy, define ∇(x,y)=x−1y.\nabla(x,y)=x-\frac1y .∇(x,y)=x−y1. What is the value of ∇(2,∇(2,∇(2,…∇(2,∇(2,2))…)))⏟2016 ∇s?\underbrace{\nabla(2, \nabla(2, \…
- Let aaa and bbb be real numbers, and let f(x)=ax+bf(x)=ax+bf(x)=ax+b. Find all pairs (a,b)(a,b)(a,b) of real numbers such that f(f(x))=4x+3f(f(x))=4x+3f(f(x))=4x+3 for all real …
- Suppose that f(x)=kx2x−1f(x) = \frac{kx}{2x-1}f(x)=2x−1kx for x≠12x \neq \frac12x=21 satisfies f(f(x))=xf(f(x)) = xf(f(x))=x for all real numbers xxx except 12\frac1221. Wha…
- Let f(x)=5x+3f(x) = 5x + 3f(x)=5x+3 and g(x)=2x−4g(x) = 2x - 4g(x)=2x−4. What is f(g(2))f(g(2))f(g(2))? Note: This is also written as (f∘g)(2)(f \circ g)(2)(f∘g)(2).
- Let f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x+2.g(x) = x + 2.g(x)=x+2. What is f(g(x))f(g(x))f(g(x))?
- Let f(x)=2xf(x) = 2^xf(x)=2x and g(x)=14x2g(x) = \frac{1}{4}x^2g(x)=41x2. What is f(g(4))−f(g(2))f(g(4)) - f(g(2))f(g(4))−f(g(2))?
- For a function fff, we can denote f∘n(x)=f(f(f(f(…f(x)… ))))f^{\circ n}(x) = f(f(f(f(\dots f(x)\dots))))f∘n(x)=f(f(f(f(…f(x)…)))), applying fff a total of nnn times. This means …
- If f(x)=1xf(x) = \frac{1}{x}f(x)=x1, what is f(f(x))f(f(x))f(f(x))?
- True or False\text{True or False}True or False === If (f∘f)(x)=x(f \circ f)(x) = x(f∘f)(x)=x, then either f(x)=xf(x) = xf(x)=x or f(x)=−xf(x) = -xf(x)=−x.
- Let f(x)=x2f(x) = x^2f(x)=x2, g(x)=x+2g(x) = x+2g(x)=x+2, and h(x)=3xh(x) = 3xh(x)=3x. Which of the following functions can NOT be represented as a composition of two of these f…
- Given g(x)=3x−xg(x)=3x-\sqrt xg(x)=3x−x, evaluate g(t−2)g(t-2)g(t−2)
- Let f(x)=x3+3xf(x)=x^3+3xf(x)=x3+3x and g(x)=xg(x)=\sqrt xg(x)=x, find f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x)).
- If f(x)=4x+8f(x)=4x+8f(x)=4x+8 and g(x)=7−x2,g(x)=7-x^2,g(x)=7−x2, calculate f(g(0))f(g(0))f(g(0)) and g(f(0)).g(f(0)).g(f(0)).
- Given f(x)=1x−6 , g(x)=7x+6f(x)=\dfrac{1}{x-6}\:,\:g(x)=\dfrac{7}{x}+6f(x)=x−61,g(x)=x7+6, find f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x))
- Given f(x)=1x−4 , g(x)=2x+4f(x)=\dfrac{1}{x-4}\:,\:g(x)=\dfrac{2}{x}+4f(x)=x−41,g(x)=x2+4, find f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x))
- Given f(x)=x2+1 , g(x)=x+2f(x)=x^2+1\:,\:g(x)=\sqrt{x+2}f(x)=x2+1,g(x)=x+2, find f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x))
- Given f(x)=x+2 , g(x)=x2+3f(x)=\sqrt x+2\:,\:g(x)=x^2+3f(x)=x+2,g(x)=x2+3, find f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x))
- Given f(x)=∣x∣ , g(x)=5x+1f(x)=|x|\:,\:g(x)=5x+1f(x)=∣x∣,g(x)=5x+1, find f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x))
- Given f(x)=x3 , g(x)=x2+3f(x)=\sqrt[3]x\:,\:g(x)=x^2+3f(x)=3x,g(x)=x2+3, find f(g(x))f(g(x))f(g(x)) and g(f(x))g(f(x))g(f(x))
- If f(x)=x4+6f(x)=x^4+6f(x)=x4+6, g(x)=x−6g(x)= x-6g(x)=x−6, and h(x)=xh(x)=\sqrt xh(x)=x, find f(g(h(x)))f(g(h(x)))f(g(h(x)))
- If f(x)=x2+1f(x)=x^2+1f(x)=x2+1, g(x)=1xg(x)= \dfrac{1}{x}g(x)=x1, and h(x)=x+3h(x)=x+3h(x)=x+3, find f(g(h(x)))f(g(h(x)))f(g(h(x)))
- Given functions p(x)=1xp(x)=\dfrac1{\sqrt x}p(x)=x1 and m(x)=x2−4m(x)=x^2-4m(x)=x2−4, what is the domain of p(x)m(x)?\dfrac{p(x)}{m(x)}?m(x)p(x)?
- Given functions p(x)=1xp(x)=\dfrac1{\sqrt x}p(x)=x1 and m(x)=x2−4m(x)=x^2-4m(x)=x2−4, what is the domain of p(m(x)) p(m(x))\:p(m(x))?
- Given functions p(x)=1xp(x)=\dfrac1{\sqrt x}p(x)=x1 and m(x)=x2−4m(x)=x^2-4m(x)=x2−4, what is the domain of m(p(x)) m(p(x))\:m(p(x))?
- Given functions q(x)=1xq(x)=\dfrac1{\sqrt x}q(x)=x1 and h(x)=x2−9h(x)=x^2-9h(x)=x2−9, what is the domain of h(q(x)) h(q(x))\:h(q(x))?
- Given functions q(x)=1xq(x)=\dfrac1{\sqrt x}q(x)=x1 and h(x)=x2−9h(x)=x^2-9h(x)=x2−9, what is the domain of q(x)h(x) \dfrac{q(x)}{h(x)}\:h(x)q(x)?
- Given functions q(x)=1xq(x)=\dfrac1{\sqrt x}q(x)=x1 and h(x)=x2−9h(x)=x^2-9h(x)=x2−9, what is the domain of q(h(x)) q(h(x))\:q(h(x))?
- The function D(p)D(p)D(p) gives the number of items that will be demanded when the price is ppp. The production cost, C(x)C(x)C(x) is the cost of producing xxx items. To determi…
- The function A(d)A(d)A(d) gives the pain level on a scale of 0−100-100−10 experienced by a patient with ddd milligrams of pain reduction drug in their system. The milligrams of …
- The radius rrr, in inches, of a spherical balloon is related to the volume, VVV, by r(V)=3V4π3r(V)=\sqrt[3]{\dfrac{3V}{4\pi }}r(V)=34π3V. Air is pumped into the balloon, so th…
- The number of bacteria in a refrigerated food product is given by N(T)=23T2−56T+1N(T)=23T^2-56T+1N(T)=23T2−56T+1, 3<T<333<T<333<T<33, where TTT is the temperature of the food. W…
- The number of bacteria in a refrigerated food product is given by N(T)=23T2−56T+1N(T)=23T^2-56T+1N(T)=23T2−56T+1, 3<T<333<T<333<T<33, where TTT is the temperature of the food. W…
- The radius rrr, in inches, of a spherical balloon is related to the volume, VVV, by r(V)=3V4π3r(V)=\sqrt[3]{\dfrac{3V}{4\pi }}r(V)=34π3V. Air is pumped into the balloon, so th…
- Find functions f(x)f(x)f(x) and g(x)g(x)g(x) such that f(g(x))=(x+2)2f(g(x))=(x+2)^2f(g(x))=(x+2)2.
- Find functions f(x)f(x)f(x) and g(x)g(x)g(x) such that f(g(x))=(x−5)3.f(g(x))=(x-5)^3.f(g(x))=(x−5)3.
- Find functions f(x)f(x)f(x) and g(x)g(x)g(x) such that f(g(x))=3x−5.f(g(x))=\dfrac{3}{x-5}.f(g(x))=x−53.
- Find functions f(x)f(x)f(x) and g(x)g(x)g(x) such that f(g(x))=4(x+2)2.f(g(x))=\dfrac{4}{(x+2)^2}.f(g(x))=(x+2)24.
- Find functions f(x)f(x)f(x) and g(x)g(x)g(x) such that f(g(x))=4+x3.f(g(x))=4+\sqrt[3]{x}.f(g(x))=4+3x.
- Find functions f(x)f(x)f(x) and g(x)g(x)g(x) such that f(g(x))=3+x−2.f(g(x))=3+\sqrt{x-2}.f(g(x))=3+x−2.
- Let g(x)g(x)g(x) be a linear function, with the form g(x)=ax+bg(x)=ax+bg(x)=ax+b for constants aaa and bbb. Find a function g(x)g(x)g(x) such that g(g(x))=6x−8g(g(x))=6x-8g(g(x)…
- True or False\text{True or False}True or False === (f∘g)(x)=(g∘f)(x)(f\circ g)(x)=(g\circ f)(x)(f∘g)(x)=(g∘f)(x) always.
- True or False\text{True or False}True or False === It is always true that (f∘g)(x)+c=[(f+c)∘(g+c)](x).(f\circ g)(x)+c=[(f+c)\circ (g+c)](x).(f∘g)(x)+c=[(f+c)∘(g+c)](x).
- If f(x)=3x2+4x−9f(x)=3x^2+4x-9f(x)=3x2+4x−9 and g(x)=3x−4g(x)=3x-4g(x)=3x−4 find (f∘g)(x)(f\circ g)(x)(f∘g)(x).
- If f(x)=3x2+4x−9f(x)=3x^2+4x-9f(x)=3x2+4x−9 and g(x)=3x−4g(x)=3x-4g(x)=3x−4 find (g∘f)(x)(g\circ f)(x)(g∘f)(x).
- If f(x)=x2−7f(x)=x^2-7f(x)=x2−7 and g(x)=ln(x)−3x2ex−x7g(x)=\dfrac{\ln(x)-3^x}{2e^x-x^7}g(x)=2ex−x7ln(x)−3x find (f∘f)(x)(f\circ f)(x)(f∘f)(x).
- Let f(x)=x3,g(x)=ex2,f(x)=x^3, g(x)=e^{x^2},f(x)=x3,g(x)=ex2, and h(x)=lnx.h(x)=\ln x.h(x)=lnx. What is h(g(f(x)))?h(g(f(x)))?h(g(f(x)))?
- Let f(x)=x3,g(x)=ex2,f(x)=x^3, g(x)=e^{x^2},f(x)=x3,g(x)=ex2, and h(x)=lnx.h(x)=\ln x.h(x)=lnx. What is f(g(h(x)))?f(g(h(x)))?f(g(h(x)))?
- Let f(x)=x3,g(x)=ex2,f(x)=x^3, g(x)=e^{x^2},f(x)=x3,g(x)=ex2, and h(x)=lnx.h(x)=\ln x.h(x)=lnx. What is g(f(h(x)))?g(f(h(x)))?g(f(h(x)))?
- Let f(x)=exf(x)=e^xf(x)=ex and g(x)=lnx.g(x)=\ln x.g(x)=lnx. What are f(g(x))f(g(x))f(g(x)) and g(f(x))?g(f(x))?g(f(x))?
- Let f(x)=x2.f(x)=x^2.f(x)=x2. What is f(f(f(x)))?f(f(f(x)))?f(f(f(x)))?
- Let f(x)=lnx.f(x)=\ln x.f(x)=lnx. Find f(f(1)).f(f(1)).f(f(1)).
- If limx→ag(x)=5\lim_{x\to{a}} g(x)=5limx→ag(x)=5, and f(x)=x+1f(x)=x+1f(x)=x+1, what is the limx→af(g(x))\lim_{x\to{a}} f(g(x))limx→af(g(x))?
- If f(x)=x3+x2f(x) = x^3 + x^2f(x)=x3+x2 and g(x)=x2,g(x) = x^2,g(x)=x2, what is f(g(x))?f(g(x))?f(g(x))?
- If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x2+1,g(x) = x^2 + 1,g(x)=x2+1, for what real value(s) of xxx does f2(g(x))=g(f2(x))?f^2(g(x)) = g(f^2(x))?f2(g(x))=g(f2(x))?
- Let f(x)=sin(x)f(x) = \sin(x)f(x)=sin(x) and g(x)=x2+π.g(x) = x^2 + \pi.g(x)=x2+π. What is f(g(x))?f(g(x))?f(g(x))?
- Let f(x)=sin(x)f(x) = \sin(x)f(x)=sin(x) and g(x)=cos(x).g(x) = \cos(x).g(x)=cos(x). Find f(g(x)).f(g(x)).f(g(x)).
- Let f(x)=xsin(x)f(x) = x\sin(x)f(x)=xsin(x) and g(x)=x2.g(x) = x^2.g(x)=x2. Find g(f(x)).g(f(x)).g(f(x)).
- Let f(x)=x3f(x) = x^3f(x)=x3 and g(x)=x5,g(x) = x^5,g(x)=x5, then what is g(f(x))?g(f(x))?g(f(x))?