https://peda.net/id/c4ca39f4166 2019-12-04T10:12:09+02:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/07948296203 2019-12-16T22:02:47+02:00 a) <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(1%5Cright)%3D3" alt="f\left(1\right)=3"/><br/> b) <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(4%5Cright)%3D2" alt="f\left(4\right)=2"/><br/> c) <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(f%5Cleft(1%5Cright)%5Cright)%3D2" alt="g\left(f\left(1\right)\right)=2"/><br/> d) <img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(f%5Cleft(2%5Cright)%5Cright)%3D4" alt="g\left(f\left(2\right)\right)=4"/><br/> e) nollakohta on <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D4" alt="x=4"/> 2019-12-16T22:02:47+02:00 https://peda.net/id/b026b9d4167 2019-12-04T11:44:38+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Ccirc%20g%3D3%5Ccos%5Cleft(x%5Cright)" alt="f\circ g=3\cos\left(x\right)"/></div> <div>arvojoukko <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7B%5D%7D-3%7B%2C%7D3%5Ctext%7B%5B%7D" alt="\text{]}-3{,}3\text{[}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Ccirc%20f%3D%5Ccos%5Cleft(3x%5Cright)" alt="g\circ f=\cos\left(3x\right)"/></div> <div>arvojoukko <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctext%7B%5D%7D-1%7B%2C%7D1%5Ctext%7B%5B%7D" alt="\text{]}-1{,}1\text{[}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(g%5Cleft(x%5Cright)%5Cright)%3D2" alt="f\left(g\left(x\right)\right)=2"/> on ratkaisuja, koska 2 on arvojoukossa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(f%5Cleft(x%5Cright)%5Cright)%3D2" alt="g\left(f\left(x\right)\right)=2"/> ei ole ratkaisuja, koska 2 on arvojoukon ulkopuolella</div> <div> </div> 2019-12-04T11:44:38+02:00 https://peda.net/id/a095556c167 2019-12-04T11:37:02+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Csin%20x" alt="u\left(x\right)=\sin x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D%5Cfrac%7Bx%7D%7B4%7D" alt="s\left(x\right)=\frac{x}{4}"/><br/> b)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E3" alt="u\left(x\right)=x^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D-x%2B5" alt="s\left(x\right)=-x+5"/> 2019-12-04T11:37:02+02:00 https://peda.net/id/68b9de88167 2019-12-04T11:35:29+02:00 <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Csin3x" alt="u\left(s\left(x\right)\right)=\sin3x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(%5Cpi%5Cright)%5Cright)%3D0" alt="u\left(s\left(\pi\right)\right)=0"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Csin%5Cfrac%7Bx%7D%7B3%7D" alt="u\left(s\left(x\right)\right)=\sin\frac{x}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(%5Cpi%5Cright)%5Cright)%3D%5Csin%5Cfrac%7B%5Cpi%7D%7B3%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D" alt="u\left(s\left(\pi\right)\right)=\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Csin%5Cleft(x%2B%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)" alt="u\left(s\left(x\right)\right)=\sin\left(x+\frac{\pi}{2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(%5Cpi%5Cright)%5Cright)%3D%5Csin%5Cfrac%7B3%5Cpi%7D%7B2%7D%3D-1" alt="u\left(s\left(\pi\right)\right)=\sin\frac{3\pi}{2}=-1"/></div> 2019-12-04T11:35:29+02:00 https://peda.net/id/1623d156167 2019-12-04T11:33:10+02:00 a)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E5" alt="u\left(x\right)=x^5"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D4x%2B1" alt="s\left(x\right)=4x+1"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Ctan%20x" alt="u\left(x\right)=\tan x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D2x-1" alt="s\left(x\right)=2x-1"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Ccos%20x" alt="u\left(x\right)=\cos x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3Dx%5E%7B10%7D" alt="s\left(x\right)=x^{10}"/></div> <div>d)<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E%7B10%7D" alt="u\left(x\right)=x^{10}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D%5Ccos%20x" alt="s\left(x\right)=\cos x"/></div> 2019-12-04T11:33:10+02:00 https://peda.net/id/b48df00c167 2019-12-04T11:30:26+02:00 a)<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Cleft(3x%2B2%5Cright)%5E5" alt="u\left(s\left(x\right)\right)=\left(3x+2\right)^5"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Ctan%5Cleft(4x%2B5%5Cright)" alt="u\left(s\left(x\right)\right)=\tan\left(4x+5\right)"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Csin%5E4x" alt="u\left(s\left(x\right)\right)=\sin^4x"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Csin%20x%5E4" alt="u\left(s\left(x\right)\right)=\sin x^4"/></div> 2019-12-04T11:30:26+02:00 https://peda.net/id/ffc75ce0167 2019-12-04T11:26:50+02:00 <img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3Dx%5E2%2B2%5Csin%20x" alt="f\left(x\right)=x^2+2\sin x"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=h%5Cleft(x%5Cright)%3D%5Cleft(2x%2B3%5Cright)%5E2%2B2%5Csin%5Cleft(2x%2B3%5Cright)" alt="h\left(x\right)=\left(2x+3\right)^2+2\sin\left(2x+3\right)"/></div> <div>määritelmä</div> <div>lauseke<img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)" alt="u\left(s\left(x\right)\right)"/>on funktioiden <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%5C%20ja%5C%20s%5Cleft(x%5Cright)" alt="u\left(x\right)\ ja\ s\left(x\right)"/> yhdistetty funktio</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)" alt="u\left(x\right)"/> on ulkofunktio ja <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)" alt="s\left(x\right)"/> sisäfunktio</div> <div>Merkitään <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(s%5Cleft(x%5Cright)%5Cright)%3D%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)" alt="u\left(s\left(x\right)\right)=\left(u\circ s\right)\left(x\right)"/><br/>  (luetaan uu pallo äs)</div> <div> </div> <div>esimerkki</div> <div>muodosta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)%5C%20ja%5C%20%5Cleft(s%5Ccirc%20u%5Cright)%5Cleft(x%5Cright)" alt="\left(u\circ s\right)\left(x\right)\ ja\ \left(s\circ u\right)\left(x\right)"/>, kun</div> <div>a)</div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E2%5C%20ja%5C%20s%5Cleft(x%5Cright)%3D3x%2B1" alt="u\left(x\right)=x^2\ ja\ s\left(x\right)=3x+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)%3D%5Cleft(3x%2B1%5Cright)%5E2" alt="\left(u\circ s\right)\left(x\right)=\left(3x+1\right)^2"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(s%5Ccirc%20u%5Cright)%5Cleft(x%5Cright)%3D3x%5E2%2B1" alt="\left(s\circ u\right)\left(x\right)=3x^2+1"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Ccos%20x%5C%20ja%5C%20s%5Cleft(x%5Cright)%3Dx%5E2%2B1" alt="u\left(x\right)=\cos x\ ja\ s\left(x\right)=x^2+1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)%3D%5Ccos%5Cleft(x%5E2%2B1%5Cright)" alt="\left(u\circ s\right)\left(x\right)=\cos\left(x^2+1\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(s%5Ccirc%20u%5Cright)%5Cleft(x%5Cright)%3D%5Ccos%5E2x%2B1" alt="\left(s\circ u\right)\left(x\right)=\cos^2x+1"/></div> <div> </div> <div>huom! yleensä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(u%5Ccirc%20s%5Cright)%5Cleft(x%5Cright)%5Cne%5Cleft(s%5Ccirc%20u%5Cright)%5Cleft(x%5Cright)" alt="\left(u\circ s\right)\left(x\right)\ne\left(s\circ u\right)\left(x\right)"/><br/> <br/> <br/> <div>tulkitse yhdistetyksi funktioksi</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%5Cleft(x%5Cright)%3D%5Cleft(3x%5E2%2B2x%5Cright)%5E2" alt="f\left(x\right)=\left(3x^2+2x\right)^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3Dx%5E2" alt="u\left(x\right)=x^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D3x%5E2%2B2x" alt="s\left(x\right)=3x^2+2x"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cleft(x%5Cright)%3D%5Cfrac%7B3%7D%7B1%2B%5Csin%20x%7D" alt="g\left(x\right)=\frac{3}{1+\sin x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=u%5Cleft(x%5Cright)%3D%5Cfrac%7B3%7D%7Bx%7D" alt="u\left(x\right)=\frac{3}{x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%5Cleft(x%5Cright)%3D1%2B%5Csin%20x" alt="s\left(x\right)=1+\sin x"/></div> </div> 2019-12-04T11:18:14+02:00