Question about homestead exemptions and property taxes

We bought a new house last year (June 2021), and the 2021 tax appraisal was $X. I just got a letter stating that the 2022 appraisal is a whopping 40% higher. I filed for a homestead exemption earlier this year as well, but the appraisal notice is showing that there are no active exemptions on the house. Does that mean I did something wrong with the filing?

Am I doing something wrong or misunderstanding homestead exemptions? I thought I should be able to get a homestead exemption for this year and cap the property tax increase at 10%.

Thank you — we're new to homeownership, property taxes, exemptions, etc.