Hola alguien que me pueda ayudar con " Como hallar el rango de esta funcíonf(x) = (X + 16) / √(x - 17)?

Función <img src="https://tex.z-dn.net/?f=f%28x%29%3D%5Cdfrac%7Bx%2B16%7D%7B%5Csqrt%7Bx-17%7D%7D" />

Es fácil ver que<img src="https://tex.z-dn.net/?f=%5Ctext%7BDom%20%7Df%20%3D%20%2817%2C%2B%5Cinfty%29" /> y que en ese dominio la función es biyectiva, por eso tiene inversa.

Tratemos de despejar <img src="https://tex.z-dn.net/?f=x" /> en términos de <img src="https://tex.z-dn.net/?f=y%3Df%28x%29" /> <img src="https://tex.z-dn.net/?f=y%3D%5Cdfrac%7Bx%2B16%7D%7B%5Csqrt%7Bx-17%7D%7D%5C%5C%20%5C%5C%20%5C%5C%0Ay%5E2%3D%5Cdfrac%7B%28x%2B16%29%5E2%7D%7Bx-17%7D%20%5C%5C%20%5C%5C%20%5C%5C%0A%28x-17%29y%5E2%3Dx%5E2%2B32x%2B256%5C%5C%20%5C%5C%20%5C%5C%0Ax%5E2%2B%2832-y%5E2%29x%2B%2817y%5E2%2B256%29%3D0%5C%5C%20%5C%5C%20%5C%5C%0Ax%3D%5Cdfrac%7By%5E2-32%5Cpm%5Csqrt%7B%28y%5E2-32%29%5E2-4%2817y%5E2%2B256%29%7D%7D%7B2%7D%5C%5C%20%5C%5C%20%5C%5C%0Ax%3D%5Cdfrac%7By%5E2-32%5Cpm%5Csqrt%7By%5E4-132y%5E2%7D%7D%7B2%7D%5C%5C%20%5C%5C%20%5C%5C%0Ax%3D%5Cdfrac%7By%5E2-32%5Cpm%7Cy%7C%5Csqrt%7By%5E2-132%7D%7D%7B2%7D%5C%5C%20%5C%5C%20%5C%5C" />

como<img src="https://tex.z-dn.net/?f=f%3A%2817%2C%2B%5Cinfty%29%5Csubset%5Cmathbb%20R%5Cto%20%5Cmathbb%20R" /> entonces

<img src="https://tex.z-dn.net/?f=y%5E2-132%3Df%5E2%28x%29-132%5Cgeq%200%5Ciff%20%5Cleft%28f%28x%29-%5Csqrt%7B132%7D%5Cright%29%5Cleft%28f%28x%29%2B%5Csqrt%7B132%7D%5Cright%29%5Cgeq%200%5C%5C%20%5C%5C%0A%5Cleft%28f%28x%29-2%5Csqrt%7B33%7D%5Cright%29%5Cleft%28f%28x%29%2B2%5Csqrt%7B33%7D%5Cright%29%5Cgeq%200%5C%5C%20%5C%5C%0Af%28x%29%5Cin%20%5Cleft%28-%5Cinfty%2C-2%5Csqrt%7B33%7D~%5Cright%5D%5Ccup%5B~2%5Csqrt%7B33%7D%2C%2B%5Cinfty%29%5C%5C%20%5C%5C%20%5C%5C%0A~~~~~~~~~~~~~~~~%5Ctext%7BRan%20%7Df%3D%5Cleft%28-%5Cinfty%2C-2%5Csqrt%7B33%7D~%5Cright%5D%5Ccup%5B~2%5Csqrt%7B33%7D%2C%2B%5Cinfty%29%0A" />.