Question

os gráficos das funções reais definidas por f(x) = x² - 1 e g(x) = k elevado a x,1≠k>0, se interceptam num ponto de abcissa 3. então o valor de f(g(k)) é

Answer 1

$\mathrm{f(x)=x^2-1\ \ \|\ \ g(x)=k^x\ \to\ f\cap g=(3,y)}\\\\ \mathbf{Calculando\ y:}\\\\ \mathrm{f(3)=3^2-1=9-1=8\ \to\ f\cap g=(3,8)}\\\\ \mathbf{Calculando\ k:}\\\\ \mathrm{f(3)=k^3\ \to\ 8=k^3\ \to\ k=\sqrt[3]{8}\ \to\ k=2}\\\\ \mathbf{Calculando\ f(g(x)):}\\\\ \mathrm{f(g(x))={[g(x)]}^2-1=\big(2^x\big)^2-1=2^{2x}-1}\\\\ \mathbf{Calculando\ f(g(k)):}\\\\ \mathrm{f(g(k))=f(g(2))=2^{2.2}-1=2^4-1=16-1}\\\\ \boxed{\mathbf{f(g(k))=15}}$