Find Number of Onto Functions from \(A\) to \(B\) ๐ Question Let: \[ A=\{1,2,3,4\}, \quad B=\{a,b\} \] Find the total number of onto (surjective) functions from \(A\) to \(B\). โ Solution ๐น Step 1: Total number of functions \[ \text{Total functions} = 2^4 = 16 \] — ๐น Step 2: Subtract non-onto functions Non-onto functions […]
Evaluate \((f^{-1}\circ f)(1)+\cdots+(f^{-1}\circ f)(100)\) ๐ Question Let \(f\) be an invertible real function. Evaluate: \[ (f^{-1}\circ f)(1) +(f^{-1}\circ f)(2)+ \cdots +(f^{-1}\circ f)(100) \] โ Solution ๐น Step 1: Use identity property For an invertible function: \[ (f^{-1}\circ f)(x)=x \] — ๐น Step 2: Apply to each term \[ (f^{-1}\circ f)(1)=1,\quad (f^{-1}\circ f)(2)=2,\ \ldots,\ (f^{-1}\circ f)(100)=100
Find \((f\circ f^{-1})(1)\) ๐ Question Let: \[ f:\mathbb{R}\to\mathbb{R}, \quad f(x)=\frac{2x-3}{4} \] Find \((f\circ f^{-1})(1)\). โ Solution ๐น Step 1: Key Property If \(f\) is invertible, then: \[ (f\circ f^{-1})(x)=x \] — ๐น Step 2: Apply property \[ (f\circ f^{-1})(1)=1 \] — ๐ฏ Final Answer \[ \boxed{1} \] ๐ Exam Shortcut \(f(f^{-1}(x))=x\) No need to find
Let f:RโR be defined as f(x) = (2x-3)/4. Write fof^-1(1) Read More ยป
Find \((f\circ g)(-2)\) ๐ Question Let: \[ f(x)=x^2+x+1,\quad g(x)=1-x^2 \] Find \((f\circ g)(-2)\). โ Solution ๐น Step 1: Use definition \[ (f\circ g)(x)=f(g(x)) \] — ๐น Step 2: Find \(g(-2)\) \[ g(-2)=1-(-2)^2=1-4=-3 \] — ๐น Step 3: Find \(f(g(-2))\) \[ f(-3)=(-3)^2+(-3)+1 \] :contentReference[oaicite:0]{index=0} — ๐ฏ Final Answer \[ \boxed{(f\circ g)(-2)=7} \] ๐ Exam Shortcut Compute
Find \(f^{-1}(x)\) and Range of \(f(x)=\frac{2x}{5x+3}\) ๐ Question Let: \[ f:\mathbb{R}\setminus\left\{-\frac{3}{5}\right\}\to \mathbb{R}, \quad f(x)=\frac{2x}{5x+3} \] Find \(f^{-1}(x)\) and the range of \(f\). โ Solution ๐น Step 1: Find inverse Let: \[ y=\frac{2x}{5x+3} \] Cross multiply: \[ y(5x+3)=2x \] \[ 5xy+3y=2x \] \[ 3y=x(2-5y) \] \[ x=\frac{3y}{2-5y} \] Interchange \(x\) and \(y\): :contentReference[oaicite:0]{index=0} — ๐น Step
Find \(f^{-1}(x)\) for \(f(x)=\frac{x}{x+1}\) ๐ Question Let: \[ f:\mathbb{R}\setminus\{-1\}\to \mathbb{R}\setminus\{1\}, \quad f(x)=\frac{x}{x+1} \] Find \(f^{-1}(x)\). โ Solution ๐น Step 1: Let \[ y=\frac{x}{x+1} \] — ๐น Step 2: Solve for \(x\) \[ y(x+1)=x \] \[ yx+y=x \] \[ y=x-xy \] \[ y=x(1-y) \] \[ x=\frac{y}{1-y} \] — ๐น Step 3: Write inverse Interchange \(x\) and
Let f:R-{-1}โR-{1} be given by f(x) = x/ x+1. Write f^-1(x) Read More ยป
Find \(f^{-1}(x)\) for \(f(x)=a^x\) ๐ Question Let: \[ f:\mathbb{R}\to \mathbb{R}^+, \quad f(x)=a^x,\quad a>0,\ a\ne1 \] Find \(f^{-1}(x)\). โ Solution ๐น Step 1: Check invertibility The function \(a^x\) is strictly monotonic (increasing if \(a>1\), decreasing if \(0
Let f:RโR+ be defined by f(x) = a^x, a > 0 and aโ 1. Write f^-1(x) Read More ยป
Find Set \(A\) for Bijective Function \(f(x)=\sin x\) ๐ Question Let: \[ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to A,\quad f(x)=\sin x \] If \(f\) is bijective, find the set \(A\). โ Solution ๐น Step 1: Check injectivity \(\sin x\) is strictly increasing on \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\). Hence, \(f\) is one-one. — ๐น Step 2: Find range We know: \[ \sin\left(-\frac{\pi}{2}\right)=-1,\quad \sin\left(\frac{\pi}{2}\right)=1 \]
If f:(-ฯ/2 , ฯ/2)โA be defined by f(x)= sin x. If f is a bijection, write set A Read More ยป
Find Range of \(f(x)=\frac{|x|}{x}\) ๐ Question Let: \[ A=\{x\in \mathbb{R}:-4\le x\le 4,\ x\ne 0\} \] \[ f:A\to \mathbb{R}, \quad f(x)=\frac{|x|}{x} \] Find the range of \(f\). โ Solution ๐น Step 1: Consider cases Case 1: \(x>0\) \[ |x|=x \Rightarrow f(x)=\frac{x}{x}=1 \] Case 2: \(x0\) \(\frac{|x|}{x} = -1\) if \(x
Find \((f\circ g)(-3)\) ๐ Question Let: \[ f(x)=(x+1)^2,\quad g(x)=x^2+1 \] Find \((f\circ g)(-3)\). โ Solution ๐น Step 1: Use definition of composite function \[ (f\circ g)(x)=f(g(x)) \] — ๐น Step 2: Find \(g(-3)\) \[ g(-3)=(-3)^2+1=9+1=10 \] — ๐น Step 3: Find \(f(g(-3))\) \[ f(10)=(10+1)^2=11^2 \] :contentReference[oaicite:0]{index=0} — ๐ฏ Final Answer \[ \boxed{(f\circ g)(-3)=121} \] ๐