given
f ( x ) = 2 x + 5 f(x)=2x+5 f ( x ) = 2 x + 5
l e t g ( x ) = f ( x ) = 2 x + 5 let\space g(x)=\sqrt{\smash[b]{f(x)}}=\sqrt{\smash[b]{2x+5}} l e t g ( x ) = f ( x ) = 2 x + 5
The graphs of y = f ( x ) y=f(x) y = f ( x ) and y = g ( x ) y=g(x) y = g ( x ) will intersect at points where f ( x ) = g ( x ) f(x)=g(x) f ( x ) = g ( x )
f ( x ) = g ( x ) ⇒ 2 x + 5 = 2 x + 5 f(x)=g(x)\Rightarrow 2x+5=\sqrt{\smash[b]{2x+5}} f ( x ) = g ( x ) ⇒ 2 x + 5 = 2 x + 5
⇒ 2 x + 5 − 2 x + 5 = 0 \Rightarrow2x+5-\sqrt{\smash[b]{2x+5}}=0 ⇒ 2 x + 5 − 2 x + 5 = 0
⇒ 2 x + 5 ( 2 x + 5 − 1 ) = 0 \Rightarrow\sqrt{\smash[b]{2x+5}}(\sqrt{\smash[b]{2x+5}}-1)=0 ⇒ 2 x + 5 ( 2 x + 5 − 1 ) = 0
⇒ 2 x + 5 = 0 , 2 x + 5 = 1 \Rightarrow\sqrt{\smash[b]{2x+5}}=0,\sqrt{\smash[b]{2x+5}}=1 ⇒ 2 x + 5 = 0 , 2 x + 5 = 1
⇒ 2 x + 5 = 0 , 2 x + 5 = 1 \Rightarrow2x+5=0,2x+5=1 ⇒ 2 x + 5 = 0 , 2 x + 5 = 1
⇒ x = − 5 2 , x = − 2 \Rightarrow x=-\frac{5}{2} \space ,\space x=-2 ⇒ x = − 2 5 , x = − 2
The y-coordinates are
x = − 2 ⇒ y = 2 x + 5 = 1 x=-2\Rightarrow y=\sqrt{\smash[b]{2x+5}}=1 x = − 2 ⇒ y = 2 x + 5 = 1
x = − 5 2 ⇒ y = 2 x + 5 = 0 x=-\frac{5}{2}\Rightarrow y=\sqrt{\smash[b]{2x+5}}=0 x = − 2 5 ⇒ y = 2 x + 5 = 0
Hence, the x-coordinates of the points of intersection are x = − 5 2 , − 2 x=-\frac{5}{2},-2 x = − 2 5 , − 2
Hence, the x-coordinates of the points of intersection are y = 0 , 1 y=0,1 y = 0 , 1
The points of intersections are ( − 2 , 1 ) (-2,1) ( − 2 , 1 ) and ( − 5 2 , 0 ) (-\frac{5}{2},0) ( − 2 5 , 0 )
Comments