Let f be a function from R to R and m and n real numbers such that mf(x−1)+nf(−x)=2∣x∣+1 for all real numbers. If f(−2)=5 and f(1)=1, then find the numbers m and n
Q. Let f be a function from R to R and m and n real numbers such that mf(x−1)+nf(−x)=2∣x∣+1 for all real numbers. If f(−2)=5 and f(1)=1, then find the numbers m and n
Substitute x=−2: Plug in x=−2 into the equation mf(x−1)+nf(−x)=2∣x∣+1.Calculation: mf(−3)+nf(2)=2∣−2∣+1=5.Since f(−2)=5, substitute and simplify.mf(−3)+5n=5.
Substitute x=1: Plug in x=1 into the equation mf(x−1)+nf(−x)=2∣x∣+1.Calculation: mf(0)+nf(−1)=2∣1∣+1=3.Since f(1)=1, substitute and simplify.m+nf(−1)=3.
Find f(−3) and f(−1): We now have two equations:1. mf(−3)+5n=52. m+nf(−1)=3We need to find values for f(−3) and f(−1) to solve for m and n.
Assume f is even: Assume f is an even function since it satisfies the given functional equation involving ∣x∣.Calculation: f(−x)=f(x), so f(−3)=f(3) and f(−1)=f(1).Since f(1)=1, f(−1)=1.
Substitute f(−1)=1: Substitute f(−1)=1 into equation 2.Calculation: m+n=3.
Find f(−3) or f(3): We need to find f(−3) or f(3). Assume f(3)=f(−3)=a.Substitute into equation 1: ma+5n=5.
Solve equations simultaneously: We now have:1. ma+5n=52. m+n=3Solve these equations simultaneously.
Express m in terms of n: From equation 2, express m in terms of n: m=3−n.Substitute into equation 1: (3−n)a+5n=5.
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